9: Multivariable and Vector Functions


9.1: Functions of Several Variables and Three Dimensional Space

1

Evaluate the function at the specified points.

$$f(x,y) = x+yx^{4}, \left(-1,3\right), \left(-4,2\right), \left(-1,-3\right)$$

At $$\left(-1,3\right)\text{:}$$

At $$\left(-4,2\right)\text{:}$$

At $$\left(-1,-3\right)\text{:}$$

$$2$$

$$508$$

$$-4$$

2

Sketch a contour diagram of each function. Then, decide whether its contours are predominantly lines, parabolas, ellipses, or hyperbolas.

1. $$\displaystyle z = y - 2 x^2$$
2. $$\displaystyle z = x^2 + 3 y^2$$
3. $$\displaystyle z = x^2 - 4 y^2$$
4. $$\displaystyle z = - 4 x^2$$
3

Match the surfaces with the verbal description of the level curves by placing the letter of the verbal description to the left of the number of the surface.

1. $$\displaystyle z = \sqrt{(x^2 + y^2)}$$
2. $$\displaystyle z = 2x + 3y$$
3. $$\displaystyle z = 2x^2 + 3y^2$$
4. $$\displaystyle z = x^2 + y^2$$
5. $$\displaystyle z = xy$$
6. $$\displaystyle z = \frac{1}{x-1}$$
7. $$\displaystyle z = \sqrt{(25 - x^2 - y^2)}$$
1. a collection of equally spaced concentric circles
2. a collection of unequally spaced concentric circles
3. two straight lines and a collection of hyperbolas
4. a collection of unequally spaced parallel lines
5. a collection of concentric ellipses
6. a collection of equally spaced parallel lines
4

The domain of the function $$f(x,y) = \sqrt x + \sqrt y$$ is

C

5

Find the equation of the sphere centered at $$(10, 10, 4)$$ with radius 9. Normalize your equations so that the coefficient of $$x^2$$ is 1.

= 0.

Give an equation which describes the intersection of this sphere with the plane $$z = 5\text{.}$$

= 0.

$$\left(x-10\right)^{2}+\left(y-10\right)^{2}+\left(z-4\right)^{2}-81$$

$$\left(x-10\right)^{2}+\left(y-10\right)^{2}+1-81$$

6

(A) If the positive z-axis points upward, an equation for a horizontal plane through the point $$\left(-4,1,5\right)$$ is

.

(B) An equation for the plane perpendicular to the x-axis and passing through the point $$\left(-4,1,5\right)$$ is

.

(C) An equation for the plane parallel to the xz-plane and passing through the point $$\left(-4,1,5\right)$$ is

.

$$z = 5$$

$$x = -4$$

$$y = 1$$

7

A car rental company charges a one-time application fee of 25 dollars, 45 dollars per day, and 12 cents per mile for its cars.

(a) Write a formula for the cost, $$C\text{,}$$ of renting a car as a function of the number of days, $$d\text{,}$$ and the number of miles driven, $$m\text{.}$$

$$C =$$

(b) If $$C = f(d, m)\text{,}$$ then $$f(3, 880) =$$

$$25+45d+\frac{12m}{100}$$

$$265.6$$

8

(a) Describe the set of points whose distance from the x-axis equals the distance from the yz-plane.

• A cylinder opening along the x-axis
• A cone opening along the x-axis
• A cone opening along the z-axis
• A cylinder opening along the z-axis
• A cone opening along the y-axis
• A cylinder opening along the y-axis

(b) Find the equation for the set of points whose distance from the x-axis equals the distance from the yz-plane.

• $$\displaystyle x^2 + y^2 = r^2$$
• $$\displaystyle x^2 + z^2 = r^2$$
• $$\displaystyle y^2 = x^2 + z^2$$
• $$\displaystyle y^2 + z^2 = r^2$$
• $$\displaystyle z^2 = x^2 + y^2$$
• $$\displaystyle x^2 = y^2 + z^2$$
9

For each surface, decide whether it could be a bowl, a plate, or neither. Consider a plate to be any fairly flat surface and a bowl to be anything that could hold water, assuming the positive z-axis is up.

1. $$\displaystyle x + y + z = 3$$
2. $$\displaystyle z = 1 - x^2 - y^2$$
3. $$\displaystyle z = x^2 + y^2$$
4. $$\displaystyle z = - \sqrt{ 3 - x^2 - y^2 }$$
5. $$\displaystyle z = 1$$
10

Consider the concentration, C, (in mg/liter) of a drug in the blood as a function of the amount of drug given, x, and the time since injection, t. For $$0 \leq x \leq 5$$ mg and $$t \geq 0$$ hours, we have

$C = f(x,t) = 22te^{-\left(5-x\right)t} \nonumber$

$$f(1,4) =$$

Give a practical interpretation of your answer: $$f(1, 4)$$ is

• the change in concentration of a 4 mg dose in the blood 1 hours after injection.
• the concentration of a 4 mg dose in the blood 1 hours after injection.
• the amount of a 1 mg dose in the blood 4 hours after injection.
• the change in concentration of a 1 mg dose in the blood 4 hours after injection.
• the concentration of a 1 mg dose in the blood 4 hours after injection.
• the amount of a 4 mg dose in the blood 1 hours after injection.

$$9.9031\times 10^{-6}$$

A manufacturer sells aardvark masks at a price of $280 per mask and butterfly masks at a price of$400 per mask. A quantity of a aardvark masks and b butterfly masks is sold at a total cost of $500 to the manufacturer. (a) Express the manufacturer's profit, P, as a function of a and b. $$P(a,b) =$$ dollars. (b) The curves of constant profit in the ab-plane are • circles • ellipses • parabolas • lines • hyperbolas Answer. $$280a+400b-500$$ 12 Consider the concentration, $$C\text{,}$$ in mg per liter (L), of a drug in the blood as a function of $$x\text{,}$$ the amount, in mg, of the drug given and $$t\text{,}$$ the time in hours since the injection. For $$0 \leq x \leq 4$$ and $$t \geq 0\text{,}$$ we have $$C = f(x,t) = t e^{-t(5-x)}\text{.}$$ Graph the following two single variable functions on a separate page, being sure that you can explain their significance in terms of drug concentration. (a) $$f(2,t)$$ (b) $$f(x,2.5)$$ Using your graph in (a), where is $$f(2,t)$$ a maximum? $$t =$$ a minimum? $$t =$$ Using your graph in (b), where is $$f(x,2.5)$$ a maximum? $$x =$$ a minimum? $$x =$$ Answer. 1 $$\frac{1}{5-2}$$ Answer. 2 $$0$$ Answer. 3 $$4$$ Answer. 4 $$0$$ 13 By setting one variable constant, find a plane that intersects the graph of $$z = 6x^{2}-3y^{2}+3$$ in a: (a) Parabola opening upward: the plane = (Give your answer by specifying the variable in the first answer blank and a value for it in the second.) (b) Parabola opening downward: the plane = (Give your answer by specifying the variable in the first answer blank and a value for it in the second.) (c) Pair of intersecting straight lines: the plane = (Give your answer by specifying the variable in the first answer blank and a value for it in the second.) Answer. 1 $$y$$ Answer. 2 $$0$$ Answer. 3 $$x$$ Answer. 4 $$0$$ Answer. 5 $$z$$ Answer. 6 $$3$$ 14 Find the equation of each of the following geometric objects. 1. The plane parallel to the $$xy$$-plane that passes through the point $$(-4,5,-12)\text{.}$$ 2. The plane parallel to the $$yz$$-plane that passes through the point $$(7, -2, -3)\text{.}$$ 3. The sphere centered at the point $$(2,1,3)$$ and has the point $$(-1,0,-1)$$ on its surface. 4. The sphere whose diameter has endpoints $$(-3,1,-5)$$ and $$(7,9,-1)\text{.}$$ 15 The Ideal Gas Law, $$PV = RT\text{,}$$ relates the pressure ($$P\text{,}$$ in pascals), temperature ($$T\text{,}$$ in Kelvin), and volume ($$V\text{,}$$ in cubic meters) of 1 mole of a gas ($$R = 8.314 \ \frac{\text{J} }{\text{ mol } \ \text{K} }$$ is the universal gas constant), and describes the behavior of gases that do not liquefy easily, such as oxygen and hydrogen. We can solve the ideal gas law for the volume and hence treat the volume as a function of the pressure and temperature: $V(P,T) = \frac{8.314T}{P}. \nonumber$ 1. Explain in detail what the trace of $$V$$ with $$P=1000$$ tells us about a key relationship between two quantities. 2. Explain in detail what the trace of $$V$$ with $$T=5$$ tells us. 3. Explain in detail what the level curve $$V = 0.5$$ tells us. 4. Use 2 or three additional traces in each direction to make a rough sketch of the surface over the domain of $$V$$ where $$P$$ and $$T$$ are each nonnegative. Write at least one sentence that describes the way the surface looks. 5. Based on all your work above, write a couple of sentences that describe the effects that temperature and pressure have on volume. 16 When people buy a large ticket item like a car or a house, they often take out a loan to make the purchase. The loan is paid back in monthly installments until the entire amount of the loan, plus interest, is paid. The monthly payment that the borrower has to make depends on the amount $$P$$ of money borrowed (called the principal), the duration $$t$$ of the loan in years, and the interest rate $$r\text{.}$$ For example, if we borrow$18,000 to buy a car, the monthly payment $$M$$ that we need to make to pay off the loan is given by the formula

$M(r,t) = \frac{1500r}{1-\frac{1}{\left(1+\frac{r}{12}\right)^{12t}}}. \nonumber$
1. Find the monthly payments on this loan if the interest rate is 6% and the duration of the loan is 5 years.
2. Create a table of values that illustrates the trace of $$M$$ with $$r$$ fixed at 5%. Use yearly values of $$t$$ from 2 to 6. Round payments to the nearest penny. Explain in detail in words what this trace tells us about $$M\text{.}$$
3. Create a table of values that illustrates the trace of $$M$$ with $$t$$ fixed at 3 years. Use rates from 3% to 11% in increments of 2%. Round payments to the nearest penny. Explain in detail what this trace tells us about $$M\text{.}$$
4. Consider the combinations of interest rates and durations of loans that result in a monthly payment of \$200. Solve the equation $$M(r,t) = 200$$ for $$t$$ to write the duration of the loan in terms of the interest rate. Graph this level curve and explain as best you can the relationship between $$t$$ and $$r\text{.}$$
17

Consider the function $$h$$ defined by $$h(x,y) = 8 - \sqrt{4 - x^2 - y^2}\text{.}$$

1. What is the domain of $$h\text{?}$$ (Hint: describe a set of ordered pairs in the plane by explaining their relationship relative to a key circle.)
2. The range of a function is the set of all outputs the function generates. Given that the range of the square root function $$g(t) = \sqrt{t}$$ is the set of all nonnegative real numbers, what do you think is the range of $$h\text{?}$$ Why?
3. Choose 4 different values from the range of $$h$$ and plot the corresponding level curves in the plane. What is the shape of a typical level curve?
4. Choose 5 different values of $$x$$ (including at least one negative value and zero), and sketch the corresponding traces of the function $$h\text{.}$$
5. Choose 5 different values of $$y$$ (including at least one negative value and zero), and sketch the corresponding traces of the function $$h\text{.}$$
6. Sketch an overall picture of the surface generated by $$h$$ and write at least one sentence to describe how the surface appears visually. Does the surface remind you of a familiar physical structure in nature?

9.2: Vectors

1.

For each of the following, perform the indicated computation.

(a) $$(10\,\mathit{\vec i}+7\,\mathit{\vec j}-5\,\mathit{\vec k}) - (-6\,\mathit{\vec i}+4\,\mathit{\vec j}+7\,\mathit{\vec k}) =$$

(b) $$(10\,\mathit{\vec i}+6\,\mathit{\vec j}-3\,\mathit{\vec k}) - 2(-3\,\mathit{\vec i}+10\,\mathit{\vec j}+8\,\mathit{\vec k}) =$$

$$16\,\mathit{\vec i}+3\,\mathit{\vec j}-12\,\mathit{\vec k}$$

$$16\,\mathit{\vec i}-14\,\mathit{\vec j}-19\,\mathit{\vec k}$$

2.

Find a vector $$\mathbf{a}$$ that has the same direction as $$\langle -6, 7, 6 \rangle$$ but has length $$5\text{.}$$

Answer: $$\mathbf{a} =$$

$$\left<-2.72728,3.18182,2.72728\right>$$

3.

Let $$\mathbf a = \lt -3,-4,-4>$$ and $$\mathbf b = \lt 2,2,4>\text{.}$$

Show that there are scalars s and t so that $$s \mathbf a + t \mathbf b = \lt 20,24,32>\text{.}$$

You might want to sketch the vectors to get some intuition.

$$s =$$

$$t =$$

$$-4$$

$$4$$

4.

Resolve the following vectors into components:

(a) The vector $$\vec v$$ in 2-space of length 5 pointing up at an angle of $$\pi/4$$ measured from the positive $$x$$-axis.

$$\vec v =$$ $$\vec i$$ + $$\vec j$$

(b) The vector $$\vec w$$ in 3-space of length 3 lying in the $$xz$$-plane pointing upward at an angle of $$2\pi/3$$ measured from the positive $$x$$-axis.

$$\vec v =$$ $$\vec i$$ + $$\vec j$$ + $$\vec k$$

$$5\cdot 0.707107$$

$$5\cdot 0.707107$$

$$3\cdot \left(-0.5\right)$$

$$0$$

$$3\cdot 0.866025$$

5.

Find all vectors $$\vec v$$ in 2 dimensions having $$||\vec v|| = 13$$ where the $$\,\mathit{\vec i}$$-component of $$\vec v$$ is $$5 \,\mathit{\vec i}\text{.}$$

vectors:

(If you find more than one vector, enter them in a comma-separated list.)

$$5\,\mathit{\vec i}+12\,\mathit{\vec j}, 5\,\mathit{\vec i}-12\,\mathit{\vec j}$$

6.

Which is traveling faster, a car whose velocity vector is $$26\vec i + 31\vec j\text{,}$$ or a car whose velocity vector is $$40\vec i\text{,}$$ assuming that the units are the same for both directions?

• the first car
• the second car
is the faster car.

At what speed is the faster car traveling?

speed =

$$\text{the first car}$$

$$40.4599$$

7.

Let $$\mathbf a = \langle 2, 3, 3 \rangle$$ and $$\mathbf b = \langle 0, 1, 4 \rangle\text{.}$$

Compute:

$$\mathbf a + \mathbf b$$ = (,,)

$$\mathbf a - \mathbf b$$ = (,,)

$$2\mathbf a$$ = (,,)

$$3\mathbf a + 4\mathbf b$$ = (,,)

$$|\mathbf a|$$ =

$$2$$

$$4$$

$$7$$

$$2$$

$$2$$

$$-1$$

$$4$$

$$6$$

$$6$$

$$6$$

$$13$$

$$25$$

$$4.69041575982343$$

8.

Find the length of the vectors

(a) $$3\,\mathit{\vec i}-\,\mathit{\vec j}-3\,\mathit{\vec k}\text{:}$$ length =

(b) $$-1.6\,\mathit{\vec i}+0.4\,\mathit{\vec j}-1.2\,\mathit{\vec k}\text{:}$$ length =

$$4.3589$$

$$2.03961$$

9.

For each of the following, perform the indicated operations on the vectors

$$\vec a = 5\,\mathit{\vec j}+\,\mathit{\vec k}\text{,}$$ $$\vec b = \,\mathit{\vec i}+5\,\mathit{\vec j}+\,\mathit{\vec k}\text{,}$$ $$\vec z = \,\mathit{\vec i}+4\,\mathit{\vec j}\text{.}$$

(a) $$5 \vec a + 4 \vec b =$$

(b) $$4 \vec a + 5 \vec b - 5 \vec z =$$

$$4\,\mathit{\vec i}+45\,\mathit{\vec j}+9\,\mathit{\vec k}$$

$$25\,\mathit{\vec j}+9\,\mathit{\vec k}$$

10.

Find the value(s) of $$a$$ making $$\vec v = 7 a\,\vec i - 3\, \vec j$$ parallel to $$\vec w = a^2\,\vec i + 9\, \vec j\text{.}$$

$$a =$$

(If there is more than one value of a, enter the values as a comma-separated list.)

$$0, -21$$

11.

(a) Find a unit vector from the point $$P=(1,2)$$ and toward the point $$Q=(6,14)\text{.}$$

$$\vec u =$$

(b) Find a vector of length 26 pointing in the same direction.

$$\vec v =$$

$$0.384615\,\mathit{\vec i}+0.923077\,\mathit{\vec j}$$

$$10\,\mathit{\vec i}+24\,\mathit{\vec j}$$

12.

A truck is traveling due north at $$40$$ km/hr approaching a crossroad. On a perpendicular road a police car is traveling west toward the intersection at $$35$$ km/hr. Both vehicles will reach the crossroad in exactly one hour. Find the vector currently representing the displacement of the truck with respect to the police car.

displacement $$\vec d =$$

$$-35\,\mathit{\vec i}-40\,\mathit{\vec j}$$

13.

Let $$\mathbf{v} = \langle 1, -2 \rangle\text{,}$$ $$\mathbf{u} = \langle 0, 4 \rangle\text{,}$$ and $$\mathbf{w} = \langle -5, 7 \rangle\text{.}$$

1. Determine the components of the vector $$\mathbf{u} - \mathbf{v}\text{.}$$
2. Determine the components of the vector $$2\mathbf{v} - 3\mathbf{u} \text{.}$$
3. Determine the components of the vector $$\mathbf{v} + 2\mathbf{u} - 7 \mathbf{w} \text{.}$$
4. Determine scalars $$a$$ and $$b$$ such that $$a \mathbf{v} + b\mathbf{u} = \mathbf{w}\text{.}$$ Show all of your work in finding $$a$$ and $$b\text{.}$$
14.

Let $$\mathbf{u} = \langle 2, 1 \rangle$$ and $$\mathbf{v} = \langle 1, 2 \rangle\text{.}$$

1. Determine the components and draw geometric representations of the vectors $$2\mathbf{u} \text{,}$$ $$\frac{1}{2}\mathbf{u} \text{,}$$ $$(-1)\mathbf{u}\text{,}$$ and $$(-3)\mathbf{u}$$ on the same set of axes.
2. Determine the components and draw geometric representations of the vectors $$\mathbf{u} + \mathbf{v} \text{,}$$ $$\mathbf{u} + 2\mathbf{v}\text{,}$$ and $$\mathbf{u} + 3\mathbf{v}$$ on the same set of axes.
3. Determine the components and draw geometric representations of the vectors $$\mathbf{u} - \mathbf{v}\text{,}$$ $$\mathbf{u} - 2\mathbf{v}\text{,}$$ and $$\mathbf{u} - 3\mathbf{v}$$ on the same set of axes.
4. Recall that $$\mathbf{u} - \mathbf{v} = \mathbf{u} + (-1)\mathbf{v}\text{.}$$ Sketch the vectors $$\mathbf{u}\text{,}$$ $$\mathbf{v}\text{,}$$ $$\mathbf{u}+\mathbf{v} \text{,}$$ and $$\mathbf{u}-\mathbf{v}$$ on the same set of axes. Use the “tip to tail” perspective for vector addition to explain the geometric relationship between $$\mathbf{u}\text{,}$$ $$\mathbf{v}\text{,}$$ and $$\mathbf{u} - \mathbf{v}\text{.}$$
15.

Recall that given any vector $$\mathbf{v} \text{,}$$ we can calculate its length, $$|\mathbf{v}|\text{.}$$ Also, we say that two vectors that are scalar multiples of one another are parallel.

1. Let $$\mathbf{v} = \langle 3,4 \rangle$$ in $$\mathbb{R}^2\text{.}$$ Compute $$|\mathbf{v}|\text{,}$$ and determine the components of the vector $$\mathbf{u} = \frac{1}{|v|} v\text{.}$$ What is the magnitude of the vector $$\mathbf{u}\text{?}$$ How does its direction compare to $$\mathbf{v}\text{?}$$
2. Let $$\mathbf{w} = 3\mathbf{i} - 3\mathbf{j}$$ in $$\mathbb{R}^2\text{.}$$ Determine a unit vector $$\mathbf{u}$$ in the same direction as $$\mathbf{w}\text{.}$$
3. Let $$\mathbf{v} = \langle 2, 3, 5 \rangle$$ in $$\mathbb{R}^3\text{.}$$ Compute $$|\mathbf{v}|\text{,}$$ and determine the components of the vector $$\mathbf{u} = \frac{1}{|v|} v\text{.}$$ What is the magnitude of the vector $$\mathbf{u}\text{?}$$ How does its direction compare to $$\mathbf{v}\text{?}$$
4. Let $$\mathbf{v}$$ be an arbitrary nonzero vector in $$\mathbb{R}^3\text{.}$$ Write a general formula for a unit vector that is parallel to $$\mathbf{v}\text{.}$$
16.

A force (like gravity) has both a magnitude and a direction. If two forces $$\mathbf{u}$$ and $$\mathbf{v}$$ are applied to an object at the same point, the resultant force on the object is the vector sum of the two forces. When a force is applied by a rope or a cable, we call that force tension. Vectors can be used to determine tension.

As an example, suppose a painting weighing 50 pounds is to be hung from wires attached to the frame as illustrated in Figure 9.2.10. We need to know how much tension will be on the wires to know what kind of wire to use to hang the picture. Assume the wires are attached to the frame at point $$O\text{.}$$ Let $$\mathbf{u}$$ be the vector emanating from point $$O$$ to the left and $$\mathbf{v}$$ the vector emanating from point $$O$$ to the right. Assume $$\mathbf{u}$$ makes a $$60^{\circ}$$ angle with the horizontal at point $$O$$ and $$\mathbf{v}$$ makes a $$45^{\circ}$$ angle with the horizontal at point $$O\text{.}$$ Our goal is to determine the vectors $$\mathbf{u}$$ and $$\mathbf{v}$$ in order to calculate their magnitudes.

1. Treat point $$O$$ as the origin. Use trigonometry to find the components $$u_1$$ and $$u_2$$ so that $$\mathbf{u} = u_1 \mathbf{i} + u_2 \mathbf{j} \text{.}$$ Since we don't know the magnitude of $$\vu\text{,}$$ your components will be in terms of $$|\mathbf{u}|$$ and the cosine and sine of some angle. Then find the components $$v_1$$ and $$v_2$$ so that $$\mathbf{v} = v_1 \mathbf{i} + v_2 \mathbf{j} \text{.}$$ Again, your components will be in terms of $$|\mathbf{v}|$$ and the cosine and sine of some angle.
2. The total force holding the picture up is given by $$\mathbf{u}+\mathbf{v} \text{.}$$ The force acting to pull the picture down is given by the weight of the picture. Find the force vector $$\mathbf{w}$$ acting to pull the picture down.
3. The picture will hang in equilibrium when the force acting to hold it up is equal in magnitude and opposite in direction to the force acting to pull it down. Equate these forces to find the components of the vectors $$\mathbf{u}$$ and $$\mathbf{v}\text{.}$$

9.3: Dot Product

1.

Find $$\mathbf a \cdot \mathbf b$$ if

$$\mathbf a = \langle -2, 2, -3 \rangle$$ and $$\mathbf b = \langle 4, 0, 3 \rangle$$

$$\mathbf a \cdot \mathbf b =$$

Is the angle between the vectors "acute", "obtuse" or "right"?

$$-17$$

obtuse

2.

Determine if the pairs of vectors below are "parallel", "orthogonal", or "neither".

$$\mathbf a = \langle -1, -2, 2 \rangle$$ and $$\mathbf b = \langle 4, 8, 10 \rangle$$ are

$$\mathbf a = \langle -1, -2, 2 \rangle$$ and $$\mathbf b = \langle 4, 8, -8 \rangle$$ are

$$\mathbf a = \langle -1, -2, 2 \rangle$$ and $$\mathbf b = \langle 2, 4, -5 \rangle$$ are

orthogonal

parallel

neither

3.

Perform the following operations on the vectors $$\vec{u} = \left\lt 0,5,-4\right>\text{,}$$ $$\vec{v} = \left\lt -2,0,3\right>\text{,}$$ and $$\vec{w} = \left\lt -3,0,1\right>\text{.}$$

$$\vec{u} \cdot \vec{w} =$$

$$(\vec{u} \cdot \vec{v}) \vec{u} =$$

$$((\vec{w} \cdot \vec{w}) \vec{u} ) \cdot \vec{u} =$$

$$\vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} =$$

$$-4$$

$$\left<0,-60,48\right>$$

$$410$$

$$-3$$

4.

Find $${ \mathbf a \cdot \mathbf b }$$ if $$\left| {\mathbf a} \right|$$ = 7, $$\left| {\mathbf b} \right|$$ = 7, and the angle between $${\mathbf a}$$ and $${\mathbf b}$$ is $$- \frac{\pi}{10}$$ radians.

$${ \mathbf a \cdot \mathbf b }$$ =

$$46.6017692984625$$

5.

What is the angle in radians between the vectors

$${\mathbf a}$$ = (6, -4, 9) and

$${\mathbf b}$$ = (5, -1, 6)?

$$0.249309305958578$$

6.

Find $${ \mathbf a \cdot \mathbf b }$$ if $$\left| {\mathbf a} \right|$$ = 9, $$\left| {\mathbf b} \right|$$ = 10, and the angle between $${\mathbf a}$$ and $${\mathbf b}$$ is $$- \frac{\pi}{3}$$ radians.

$${ \mathbf a \cdot \mathbf b }$$ =

$$45$$

7.

A constant force $${\mathbf F} = 9 {\mathbf i} + 3 {\mathbf j} + 6 {\mathbf k}$$ moves an object along a straight line from point $$(-5, -1, 5)$$ to point $$(-4, 1, 3)\text{.}$$

Find the work done if the distance is measured in meters and the magnitude of the force is measured in newtons.

Work: Nm

$$3$$

8.

A woman exerts a horizontal force of 1 pounds on a box as she pushes it up a ramp that is 3 feet long and inclined at an angle of 30 degrees above the horizontal.

Find the work done on the box.

Work: ft-lb

$$2.59807621135332$$

9.

If Yoda says to Luke Skywalker, “The Force be with you,” then the dot product of the Force and Luke should be:

• zero
• negative
• any real number
• positive
10.

Find the angle between the diagonal of a cube of side length 8 and the diagonal of one of its faces. The angle should be measured in radians.

$$0.615479708670387$$

11.

Let $$\mathbf{v} = \langle -2, 5 \rangle$$ in $$\mathbb{R}^2\text{,}$$ and let $$\mathbf{y} = \langle 0, 3, -2 \rangle$$ in $$\mathbb{R}^3\text{.}$$

1. Is $$\langle 2, -1 \rangle$$ perpendicular to $$\mathbf{v} \text{?}$$ Why or why not?
2. Find a unit vector $$\mathbf{u}$$ in $$\mathbb{R}^2$$ such that $$\mathbf{u}$$ is perpendicular to $$\mathbf{v}\text{.}$$ How many such vectors are there? Justify your answers.
3. Is $$\langle 2, -1, -2 \rangle$$ perpendicular to $$\mathbf{y} \text{?}$$ Why or why not?
4. Find a unit vector $$\mathbf{w}$$ in $$\mathbb{R}^3$$ such that $$\mathbf{w}$$ is perpendicular to $$\mathbf{y} \text{.}$$ How many such vectors are there?Justify your answers.
5. Let $$\mathbf{z} = \langle 2, 1, 0 \rangle\text{.}$$ Find a unit vector $$\mathbf{r}$$ in $$\mathbb{R}^3$$ such that $$\mathbf{r}$$ is perpendicular to both $$\mathbf{y}$$ and $$\mathbf{z} \text{.}$$ How many such vectors are there? Explain your process.
12.

Consider the triangle in $$\mathbb{R}^3$$ given by $$P=(3, 2, -1)\text{,}$$ $$Q=(1, -2, 4)\text{,}$$ and $$R=(4, 4, 0)\text{.}$$

1. Find the measure of each of the three angles in the triangle, accurate to $$0.01$$ degrees.
2. Choose two sides of the triangle, and call the vectors that form the sides (emanating from a common point) $$\mathbf{a}$$ and $$\mathbf{b}\text{.}$$
1. Compute $$proj_{\mathbf{v}} \mathbf{a}\text{,}$$ and $$proj_{\perp \mathbf{b}} \mathbf{a}\text{.}$$
2. Explain why $$proj_{\perp \mathbf{b}} \mathbf{a}$$ can be considered a height of triangle $$PQR\text{.}$$
3. Find the area of the given triangle.
13.

Let $$\mathbf{u}$$ and $$\mathbf{v}$$ be vectors in $$\mathbb{R}^5$$ with $$\mathbf{u} \cdot \mathbf{v} = -1\text{,}$$ $$| \mathbf{u} | = 2\text{,}$$ $$| \mathbf{v} | = 3\text{.}$$ Use the properties of the dot product to find each of the following.

1. $$\displaystyle \mathbf{u} \cdot 2\mathbf{v}$$
2. $$\displaystyle \mathbf{v} \cdot \mathbf{v}$$
3. $$\displaystyle (\mathbf{u} + \mathbf{v}) \cdot \mathbf{v}$$
4. $$\displaystyle (2\mathbf{u}+4\mathbf{v}) \cdot (\mathbf{u} - 7\mathbf{v})$$
5. $$|\mathbf{u}| |\mathbf{v}| \cos(\theta)\text{,}$$ where $$\theta$$ is the angle between $$\mathbf{u}$$ and $$\mathbf{v}$$
6. $$\displaystyle \theta$$
14.

One of the properties of the dot product is that $$(\mathbf{u}+\mathbf{v}) \cdot \mathbf{w} = (\mathbf{u} \cdot \mathbf{w}) + (\mathbf{v} \cdot \mathbf{w})\text{.}$$ That is, the dot product distributes over vector addition on the right. Here we investigate whether the dot product distributes over vector addition on the left.

1. Let $$\mathbf{u} = \langle 1,2,-1 \rangle\text{,}$$ $$\mathbf{v} = \langle 4,-3,6 \rangle\text{,}$$ and $$\mathbf{w} = \langle 4,7,2 \rangle\text{.}$$ Calculate
$\mathbf{u} \cdot (\mathbf{v} + \mathbf{w}) \ \ \text{ and } \ \ (\mathbf{u} \cdot \mathbf{v}) + (\mathbf{u} \cdot \mathbf{w}). \nonumber$

What do you notice?

2. Use the properties of the dot product to show that in general
$\mathbf{x} \cdot (\mathbf{y} + \mathbf{z}) = (\mathbf{x} \cdot \mathbf{y}) + (\mathbf{x} \cdot \mathbf{z}) \nonumber$

for any vectors $$\mathbf{x}\text{,}$$ $$\mathbf{y}\text{,}$$ and $$\mathbf{z}$$ in $$\mathbb{R}^n\text{.}$$

15.

When running a sprint, the racers may be aided or slowed by the wind. The wind assistance is a measure of the wind speed that is helping push the runners down the track. It is much easier to run a very fast race if the wind is blowing hard in the direction of the race. So that world records aren't dependent on the weather conditions, times are only recorded as record times if the wind aiding the runners is less than or equal to 2 meters per second. Wind speed for a race is recorded by a wind gauge that is set up close to the track. It is important to note, however, that weather is not always as cooperative as we might like. The wind does not always blow exactly in the direction of the track, so the gauge must account for the angle the wind makes with the track. Suppose a 4 mile per hour wind is blowing to aid runners by making a $$38^{\circ}$$ angle with the race track. Determine if any times set during such a race would qualify as records.

16.

Molecular geometry is the geometry determined by arrangements of atoms in molecules. Molecular geometry includes measurements like bond angle, bond length, and torsional angles. These attributes influence several properties of molecules, such as reactivity, color, and polarity.

As an example of the molecular geometry of a molecule, consider the methane $$\text{CH}_4$$ molecule, as illustrated in Figure 9.3.9. According to the Valence Shell Electron Repulsion (VSEPR) model, atoms that surround single different atoms do so in a way that positions them as far apart as possible. This means that the hydrogen atoms in the methane molecule arrange themselves at the vertices of a regular tetrahedron. The bond angle for methane is the angle determined by two consecutive hydrogen atoms and the central carbon atom. To determine the bond angle for methane, we can place the center carbon atom at the point $$\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2} \right)$$ and the hydrogen atoms at the points $$(0,0,0)\text{,}$$ $$(1,1,0)\text{,}$$ $$(1,0,1)\text{,}$$ and $$(0,1,1)\text{.}$$ Find the bond angle for methane to the nearest tenth of a degree.

9.4: The Cross Product

1.

If $$\mathbf{a} = \mathbf{i} + \mathbf{j} + 3 \mathbf{k}$$ and $$\mathbf{b} = \mathbf{i} + \mathbf{j} + 2 \mathbf{k}$$

Compute the cross product $$\bf{a} \times \bf{b}\text{.}$$

$$\bf{a} \times \bf{b}=$$ $$\bf{i}$$ + $$\bf{j}$$ + $$\bf{k}$$

$$-1$$

$$1$$

$$0$$

2.

Suppose $$\vec v\cdot \vec w=6$$ and $$||\vec v\times\vec w|| = 4\text{,}$$ and the angle between $$\vec v$$ and $$\vec w$$ is $$\theta\text{.}$$ Find

(a) $$\tan\theta =$$

(b) $$\theta =$$

$$\frac{4}{6}$$

$$\tan^{-1}\!\left(\frac{4}{6}\right)$$

3.

You are looking down at a map. A vector $$\bf{u}$$ with $$\left| \mathbf{u} \right|$$ = 7 points north and a vector $$\mathbf{v}$$ with $$\left| \mathbf{v} \right|$$ = 6 points northeast. The crossproduct $$\mathbf{u} \times \mathbf{v}$$ points:

A) south

B) northwest

C) up

D) down

The magnitude $$\left| \mathbf{u} \times \mathbf{v} \right|$$ =

D

$$29.698484809835$$

4.

If $$\mathbf{a} = \mathbf{i} + 8 \mathbf{j} + \mathbf{k}$$ and $$\mathbf{b} = \mathbf{i} + 10 \mathbf{j} + \mathbf{k}\text{,}$$ find a unit vector with positive first coordinate orthogonal to both $$\mathbf{a}$$ and $$\mathbf{b}\text{.}$$

$$\mathbf{i}$$ + $$\mathbf{j}$$ + $$\mathbf{k}$$

$$0.707106781186547$$

$$0$$

$$-0.707106781186547$$

5.

Sketch the triangle with vertices $$O, P=\left(0,7,6\right)$$ and $$Q=\left(7,0,2\right)$$ and compute its area using cross products.

Area=

$$\frac{1}{2}\sqrt{4361}$$

6.

Let $$A = \left(5,0,0\right)\text{,}$$ $$B = \left(2,-2,-3\right)\text{,}$$ and $$P = (k,k,k)\text{.}$$ The vector from $$A$$ to $$B$$ is perpendicular to the vector from $$A$$ to $$P$$ when $$k$$ = .

$$1.875$$

7.

Find two unit vectors orthogonal to $$\mathbf a = \langle -4, -4, 5\rangle$$ and $$\mathbf b = \langle -1, -2, 3\rangle$$

Enter your answer so that the first non-zero coordinate of the first vector is positive.

First Vector: $$\langle$$, , $$\rangle$$

Second Vector: $$\langle$$, , $$\rangle$$

$$0.240771706171538$$

$$-0.842700971600384$$

$$-0.481543412343077$$

$$-0.240771706171538$$

$$0.842700971600384$$

$$0.481543412343077$$

8.

Use the geometric definition of the cross product and the properties of the cross product to make the following calculations.

(a) $$((\vec{i} + \vec{j}) \times \vec{i} ) \times \vec{j}$$ =

(b) $$( \vec{j} + \vec{k} ) \times ( \vec{j} \times \vec{k} )$$ =

(c) $$5 \vec{i} \times ( \vec{i} + \vec{j} )$$ =

(d) $$( \vec{k} + \vec{j} ) \times ( \vec{k} - \vec{j} )$$ =

$$\left<1,0,0\right>$$

$$\left<0,1,-1\right>$$

$$\left<0,0,5\right>$$

$$\left<2,0,0\right>$$

9.

Are the following statements true or false?

1. For any scalar $$c$$ and any vector $$\vec{v}\text{,}$$ we have $$||c\vec{v}|| = c ||\vec{v}||\text{.}$$
2. The value of $$\vec{v} \cdot (\vec{v} \times \vec{w})$$ is always zero.
3. If $$\vec{v}$$ and $$\vec{w}$$ are any two vectors, then $$||\vec{v} + \vec{w}|| = ||\vec{v}|| + ||\vec{w}||\text{.}$$
4. $$(\vec{i} \times \vec{j}) \cdot \vec{k} = \vec{i} \cdot (\vec{j} \times \vec{k})\text{.}$$
10.

A bicycle pedal is pushed straight downwards by a foot with a 27 Newton force. The shaft of the pedal is 20 cm long. If the shaft is $$\pi / 3$$ radians past horizontal, what is the magnitude of the torque about the point where the shaft is attached to the bicycle? Nm

$$2.7$$

11.

Let $$\mathbf{u} = 2\mathbf{i} + \mathbf{j}$$ and $$\mathbf{v} = \mathbf{i} + 2\mathbf{j}$$ be vectors in $$\mathbb{R}^3\text{.}$$

1. Without doing any computations, find a unit vector that is orthogonal to both $$\mathbf{u}$$ and $$\mathbf{v}\text{.}$$ What does this tell you about the formula for $$\mathbf{u} \times \mathbf{v}\text{?}$$
2. Using the properties of the cross product and what you know about cross products involving the fundamental vectors $$\mathbf{i}$$ and $$\mathbf{j}\text{,}$$ compute $$\mathbf{u} \times \mathbf{v}\text{.}$$
3. Next, use the determinant version of Equation (9.4.1) to compute $$\mathbf{u} \times \mathbf{v}\text{.}$$ Write one sentence that compares your results in (a), (b), and (c).
4. Find the area of the parallelogram determined by $$\mathbf{u}$$ and $$\mathbf{v}\text{.}$$
12.

Let $$\mathbf{x} = \langle 1, 1, 1 \rangle$$ and $$\mathbf{y} = \langle 0, 3, -2 \rangle\text{.}$$

1. Are $$\mathbf{x}$$ and $$\mathbf{y}$$ orthogonal? Are $$\mathbf{x}$$ and $$\mathbf{y}$$ parallel? Clearly explain how you know, using appropriate vector products.
2. Find a unit vector that is orthogonal to both $$\mathbf{x}$$ and $$\mathbf{y}\text{.}$$
3. Express $$\mathbf{y}$$ as the sum of two vectors: one parallel to $$\mathbf{x}\text{,}$$ the other orthogonal to $$\mathbf{x}\text{.}$$
4. Determine the area of the parallelogram formed by $$\mathbf{x}$$ and $$\mathbf{y}\text{.}$$
13.

Consider the triangle in $$\mathbb{R}^3$$ formed by $$P(3, 2, -1)\text{,}$$ $$Q(1, -2, 4)\text{,}$$ and $$R(4, 4, 0)\text{.}$$

1. Find $$\overrightarrow{PQ}$$ and $$\overrightarrow{PR}\text{.}$$
2. Observe that the area of $$\triangle PQR$$ is half of the area of the parallelogram formed by $$\overrightarrow{PQ}$$ and $$\overrightarrow{PR}\text{.}$$ Hence find the area of $$\triangle PQR\text{.}$$
3. Find a unit vector that is orthogonal to the plane that contains points $$P\text{,}$$ $$Q\text{,}$$ and $$R\text{.}$$
4. Determine the measure of $$\angle PQR\text{.}$$
14.

One of the properties of the cross product is that $$(\mathbf{u}+\mathbf{v}) \times \mathbf{w} = (\mathbf{u} \times \mathbf{w}) + (\mathbf{v} \times \mathbf{w})\text{.}$$ That is, the cross product distributes over vector addition on the right. Here we investigate whether the cross product distributes over vector addition on the left.

1. Let $$\mathbf{u} = \langle 1,2,-1 \rangle\text{,}$$ $$\mathbf{v} = \langle 4,-3,6 \rangle\text{,}$$ and $$\mathbf{v} = \langle 4,7,2 \rangle\text{.}$$ Calculate
$\mathbf{u} \times (\mathbf{v} + \mathbf{w}) \ \ \text{ and } \ \ (\mathbf{u} \times \mathbf{v}) + (\mathbf{u} \times \mathbf{w}). \nonumber$

What do you notice?

2. Use the properties of the cross product to show that in general
$\mathbf{x} \times (\mathbf{y} + \mathbf{z}) = (\mathbf{x} \times \mathbf{y}) + (\mathbf{x} \times \mathbf{z}) \nonumber$

for any vectors $$\mathbf{x}\text{,}$$ $$\mathbf{y}\text{,}$$ and $$\mathbf{z}$$ in $$\mathbb{R}^3\text{.}$$

15.

Let $$\mathbf{u} = \langle u_1, u_2, u_3 \rangle\text{,}$$ $$\mathbf{v} = \langle v_1, v_2, v_3 \rangle\text{,}$$ and $$\mathbf{w} = \langle w_1, w_2, w_3 \rangle$$ be vectors in $$\mathbb{R}^3\text{.}$$ In this exercise we investigate properties of the triple scalar product $$(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}\text{.}$$

1. Show that $$(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w} = \left|\begin{array}{ccc} u_1 & u_2 & u_3 \$4pt] v_1 & v_2 & v_3 \\[4pt] w_1 & w_2 & w_3 \end{array} \right| \text{.}$$ 2. Show that $$\left|\begin{array}{ccc} u_1 & u_2 & u_3 \\[4pt] v_1 & v_2 & v_3 \\[4pt] w_1 & w_2 & w_3 \end{array} \right| = -\left|\begin{array}{ccc} v_1 & v_2 & v_3 \\[4pt] u_1 & u_2 & u_3 \\[4pt] w_1 & w_2 & w_3 \end{array} \right|\text{.}$$ Conclude that interchanging the first two rows in a $$3 \times 3$$ matrix changes the sign of the determinant. In general (although we won't show it here), interchanging any two rows in a $$3 \times 3$$ matrix changes the sign of the determinant. 3. Use the results of parts (a) and (b) to argue that \[ (\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w} = (\mathbf{w} \times \mathbf{u}) \cdot \mathbf{v} = (\mathbf{v} \times \mathbf{w}) \cdot \mathbf{u}. \nonumber$
4. Now suppose that $$\mathbf{u} \text{,}$$ $$\mathbf{v}\text{,}$$ and $$\mathbf{w}$$ do not lie in a plane when they eminate from a common initial point.
1. Given that the parallepiped determined by $$\mathbf{u}\text{,}$$ $$\mathbf{v}\text{,}$$ and $$\mathbf{w}$$ must have positive volume, what can we say about $$(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}\text{?}$$
2. Now suppose that $$\mathbf{u}\text{,}$$ $$\mathbf{v}\text{,}$$ and $$\mathbf{w}$$ all lie in the same plane. What value must $$(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}$$ have? Why?
3. Explain how (i.) and (ii.) show that if $$\mathbf{u}\text{,}$$ $$\mathbf{v}\text{,}$$ and $$\mathbf{w}$$ all eminate from the same initial point, then $$\mathbf{u}\text{,}$$ $$\mathbf{v}\text{,}$$ and $$\mathbf{w}$$ lie in the same plane if and only if $$(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w} = 0\text{.}$$ This provides a straightforward computational method for determining when three vectors are co-planar.

9.5: Lines and Planes in Space

1.

Rewrite the vector equation $$\mathbf{r} (t) = (-2 t) \mathbf{i} + (3 - 3 t) \mathbf{j} + (1+ 3 t) \mathbf{k}$$ as the corresponding parametric equations for the line.

$$x(t) =$$

$$y(t) =$$

$$z(t) =$$

$$0+\left(-2\right)t$$

$$3+\left(-3\right)t$$

$$1+3t$$

2.

Find the vector and parametric equations for the line through the point P(2, -1, -5) and parallel to the vector $$-5\mathbf i - 3\mathbf j - 3\mathbf k\text{.}$$

Vector Form: $$\mathbf r = \langle$$, , -5 $$\rangle + t \langle$$, , -3 $$\rangle$$

Parametric form (parameter t, and passing through P when t = 0):

$$x = x(t) =$$

$$y = y(t) =$$

$$z = z(t) =$$

$$2$$

$$-1$$

$$-5$$

$$-3$$

$$2+t\!\left(-5\right)$$

$$-1+t\!\left(-3\right)$$

$$-5+t\!\left(-3\right)$$

3.

Consider the line which passes through the point P(3, -5, -1), and which is parallel to the line $$x = 1 + 6t, y = 2 + 2t, z = 3 + 6t$$

Find the point of intersection of this new line with each of the coordinate planes:

xy-plane: (,, )

xz-plane: (,, )

yz-plane: (,, )

$$4$$

$$-4.66666666666667$$

$$0$$

$$18$$

$$0$$

$$14$$

$$0$$

$$-6$$

$$-4$$

4.

Find the point at which the line $$\langle 4, 2, 4 \rangle + t \langle -3, -3, -4 \rangle$$ intersects the plane $$-5 x+ 5 y - 3 z = 2\text{.}$$

(, , )

$$-2$$

$$-4$$

$$-4$$

5.

Find an equation of a plane containing the three points (5, -2, -2), (2, -5, 1), (2, -4, 3) in which the coefficient of $$x$$ is -9.

= 0.

$$-9\!\left(x-5\right)+6\!\left(y-\left(-2\right)\right)+\left(-3\right)\!\left(z-\left(-2\right)\right)$$

6.

Find an equation for the plane containing the line in the $$xy$$-plane where $$y = 1\text{,}$$ and the line in the $$x z$$-plane where $$z = 2\text{.}$$

equation:

$$z = -2y+2$$

7.

Find the angle in radians between the planes $$4 x + z = 1$$ and $$5 y + z = 1.$$

$$1.52321322351792$$

8.

A store sells CDs at one price and DVDs at another price. The figure below shows the revenue (in dollars) of the music store as a function of the number, $$c\text{,}$$ of CDs and the number, $$d\text{,}$$ of DVDs that it sells. The values of the revenue are shown on each line.

(Hint: for this problem there are many possible ways to estimate the requisite values; you should be able to find information from the figure that allows you to give an answer that is essentially exact.)

(a) What is the price of a CD? dollars

(b) What is the price of a DVD? dollars

$$10$$

$$14$$

9.

The table below gives the number of calories burned per minute for someone roller-blading, as a function of the person's weight in pounds and speed in miles per hour [from the August 28,1994, issue of Parade Magazine].

calories burned per minute

 weight$$\backslash$$speed 8 9 10 11 120 4.2 5.8 7.4 8.9 140 5.1 6.7 8.3 9.9 160 6.1 7.7 9.2 10.8 180 7 8.6 10.2 11.7 200 7.9 9.5 11.1 12.6

(a) Suppose that a 180 lb person and a 200 person both go 8 miles, the first at 9 mph and the second at 8 mph.

How many calories does the 180 lb person burn?

How many calories does the 200 lb person burn?

(b) We might also be interested in the number of calories each person burns per pound of their weight.

How many calories per pound does the 180 lb person burn?

How many calories per pound does the 200 lb person burn?

$$\frac{480\cdot 8.6}{9}$$

$$\frac{480\cdot 7.9}{8}$$

$$\frac{458.667}{180}$$

$$\frac{474}{200}$$

10.

The vector and parametric forms of a line allow us to easily describe line segments in space.

Let $$P_1 = (1,2,-1)$$ and $$P_2 = (-2,1,-2)\text{,}$$ and let $$\mathcal{L}$$ be the line in $$\mathbb{R}^3$$ through $$P_1$$ and $$P_2$$ as in Activity 9.5.2.

1. What value of the parameter $$t$$ makes $$(x(t), y(t), z(t)) = P_1\text{?}$$ What value of $$t$$ makes $$(x(t), y(t), z(t)) = P_2\text{?}$$
2. What $$t$$ values describe the line segment between the points $$P_1$$ and $$P_2\text{?}$$
3. What about the line segment (along the same line) from $$(7,4,1)$$ to $$(-8,-1,-4)\text{?}$$
4. Now, consider a segment that lies on a different line: parameterize the segment that connects point $$R=(4,-2,7)$$ to $$Q=(-11,4,27)$$ in such a way that $$t = 0$$ corresponds to point $$Q\text{,}$$ while $$t = 2$$ corresponds to $$R\text{.}$$
11.

This exercise explores key relationships between a pair of lines. Consider the following two lines: one with parametric equations $$x(s) = 4-2s\text{,}$$ $$y(s) = -2 + s\text{,}$$ $$z(s) = 1 + 3s\text{,}$$ and the other being the line through $$(-4, 2, 17)$$ in the direction $$\mathbf{v} = \langle -2, 1, 5 \rangle\text{.}$$

1. Find a direction vector for the first line, which is given in parametric form.
2. Find parametric equations for the second line, written in terms of the parameter $$t\text{.}$$
3. Show that the two lines intersect at a single point by finding the values of $$s$$ and $$t$$ that result in the same point. Then find the point of intersection.
4. Find the acute angle formed where the two lines intersect, noting that this angle will be given by the acute angle between their respective direction vectors.
5. Find an equation for the plane that contains both of the lines described in this problem.
12.

This exercise explores key relationships between a pair of planes. Consider the following two planes: one with scalar equation $$4x - 5y + z = -2\text{,}$$ and the other which passes through the points $$(1,1,1)\text{,}$$ $$(0,1,-1)\text{,}$$ and $$(4, 2, -1)\text{.}$$

1. Find a vector normal to the first plane.
2. Find a scalar equation for the second plane.
3. Find the angle between the planes, where the angle between them is defined by the angle between their respective normal vectors.
4. Find a point that lies on both planes.
5. Since these two planes do not have parallel normal vectors, the planes must intersect, and thus must intersect in a line. Observe that the line of intersection lies in both planes, and thus the direction vector of the line must be perpendicular to each of the respective normal vectors of the two planes. Find a direction vector for the line of intersection for the two planes.
6. Determine parametric equations for the line of intersection of the two planes.
13.

In this problem, we explore how we can use what we know about vectors and projections to find the distance from a point to a plane.

Let $$p$$ be the plane with equation $$z=-4x+3y+4\text{,}$$ and let $$Q = (4,-1,8)\text{.}$$

1. Show that $$Q$$ does not lie in the plane $$p\text{.}$$
2. Find a normal vector $$\mathbf{n}$$ to the plane $$p\text{.}$$
3. Find the coordinates of a point $$P$$ in $$p\text{.}$$
4. Find the components of $$\overrightarrow{PQ}\text{.}$$ Draw a picture to illustrate the objects found so far.
5. Explain why $$|comp_{\mathbf{n}} \overrightarrow{PQ}|$$ gives the distance from the point $$Q$$ to the plane $$p\text{.}$$ Find this distance.

9.6: Vector-Valued Functions

1.

Find the domain of the vector function

$\mathbf r(t) = \left \langle \ln(11 t), \sqrt{t + 10}, \frac{1}{\sqrt{12 - t}} \right \rangle \nonumber$

using interval notation.

Domain:

$$\left(0,12\right)$$

2.

Find a parametrization of the circle of radius $$6$$ in the xy-plane, centered at the origin, oriented clockwise. The point $$(6,0)$$ should correspond to $$t = 0\text{.}$$ Use $$t$$ as the parameter for all of your answers.

$$x(t) =$$

$$y(t) =$$

$$6\cos\!\left(t\right);\,-6\sin\!\left(t\right)$$

3.

Find a vector parametrization of the circle of radius $$7$$ in the xy-plane, centered at the origin, oriented clockwise so that the point $$(7,0)$$ corresponds to $$t = 0$$ and the point $$(0,-7)$$ corresponds to $$t = 1\text{.}$$

$$\vec{r}(t) =$$

$$\left<7\cos\!\left(\frac{\pi t}{2}\right),-7\sin\!\left(\frac{\pi t}{2}\right)\right>$$

4.

Find a vector parametric equation $$\vec{r}(t)$$ for the line through the points $$P = \left(4,1,1\right)$$ and $$Q = \left(9,-3,3\right)$$ for each of the given conditions on the parameter $$t\text{.}$$

(a) If $$\vec{r}(0) = \left\lt 4,1,1\right>$$ and $$\vec{r}(8) = \left\lt 9,-3,3\right>\text{,}$$ then

$$\vec{r}(t) =$$

(b) If $$\vec{r}(6) = P$$ and $$\vec{r}(10) = Q\text{,}$$ then

$$\vec{r}(t) =$$

(c) If the points $$P$$ and $$Q$$ correspond to the parameter values $$t = 0$$ and $$t = -2\text{,}$$ respectively, then

$$\vec{r}(t) =$$

$$\left(4,1,1\right)+\frac{t}{8}\!\left<5,-4,2\right>$$

$$\left(4,1,1\right)+\frac{t-6}{4}\!\left<5,-4,2\right>$$

$$\left(4,1,1\right)+\left(-\frac{t}{2}\right)\!\left<5,-4,2\right>$$

5.

Suppose parametric equations for the line segment between $$(9, -6)$$ and $$(-2, 5)$$ have the form:

$\begin{array}{r@{\,}c@{\,}l} x & = & a + bt \cr y & = & c + dt \end{array} \nonumber$

If the parametric curve starts at $$(9, -6)$$ when $$t=0$$ and ends at $$(-2, 5)$$ at $$t=1\text{,}$$ then find $$a\text{,}$$ $$b\text{,}$$ $$c\text{,}$$ and $$d\text{.}$$

$$a =$$,

$$b =$$,

$$c =$$,

$$d =$$.

$$9$$

$$-11$$

$$-6$$

$$11$$

6.

Find a parametrization of the curve $$x = -5 z^2$$ in the xz-plane. Use $$t$$ as the parameter for all of your answers.

$$x(t) =$$

$$y(t) =$$

$$z(t) =$$

$$-5t^{2};\,0;\,t$$

7.

Find parametric equations for the quarter-ellipse from $$(2,0,9)$$ to $$(0,-3,9)$$ centered at $$(0,0,9)$$ in the plane $$z = 9\text{.}$$ Use the interval $$0 \leq t \leq \pi/2\text{.}$$

$$x(t) =$$

$$y(t) =$$

$$z(t) =$$

$$2\cos\!\left(t\right);\,-3\sin\!\left(t\right);\,9$$

8.

Are the following statements true or false?

1. The line parametrized by $$x = 7, y = 5t, z = 6 + t$$ is parallel to the x-axis.
2. A parametrization of the graph of $$y = \ln(x)$$ for $$x > 0$$ is given by $$x = e^t, y = t$$ for $$-\infty \lt t \lt \infty\text{.}$$
3. The parametric curve $$x = (3t+4)^2, y = 5(3t+4)^2-9\text{,}$$ for $$0 \leq t \leq 3$$ is a line segment.
9.

Find a vector function that represents the curve of intersection of the paraboloid $$z = 5 x^2 + 5 y^2$$ and the cylinder $$y = 5 x^2\text{.}$$ Use the variable t for the parameter.

$$\mathbf r(t) = \langle t,$$, $$\rangle$$

$$5tt$$

$$5tt+125t^{4}$$

10.

A bicycle wheel has radius R. Let P be a point on the spoke of a wheel at a distance d from the center of the wheel. The wheel begins to roll to the right along the the x-axis. The curve traced out by P is given by the following parametric equations:

$$x = 15 \theta - 10 \sin(\theta)$$

$$y = 15 - 10 \cos(\theta)$$

What must we have for R and d?

R=

d =

$$15$$

$$10$$

11.

A standard parameterization for the unit circle is $$\langle \cos(t), \sin(t) \rangle\text{,}$$ for $$0 \le t \le 2\pi\text{.}$$

1. Find a vector-valued function $$\mathbf{r}$$ that describes a point traveling along the unit circle so that at time $$t=0$$ the point is at $$\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$$ and travels clockwise along the circle as $$t$$ increases.
2. Find a vector-valued function $$\mathbf{r}$$ that describes a point traveling along the unit circle so that at time $$t=0$$ the point is at $$\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$$ and travels counter-clockwise along the circle as $$t$$ increases.
3. Find a vector-valued function $$\mathbf{r}$$ that describes a point traveling along the unit circle so that at time $$t=0$$ the point is at $$\left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$$ and travels clockwise along the circle as $$t$$ increases.
4. Find a vector-valued function $$\mathbf{r}$$ that describes a point traveling along the unit circle so that at time $$t=0$$ the point is at $$(0,1)$$ and makes one complete revolution around the circle in the counter-clockwise direction on the interval $$[0,\pi]\text{.}$$
12.

Let $$a$$ and $$b$$ be positive real numbers. You have probably seen the equation $$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$$ that generates an ellipse, centered at $$(h,k)\text{,}$$ with a horizontal axis of length $$2a$$ and a vertical axis of length $$2b\text{.}$$

1. Explain why the vector function $$\mathbf{r}$$ defined by $$\mathbf{r}(t) = \langle a\cos(t), b\sin(t) \rangle\text{,}$$ $$0 \le t \le 2\pi$$ is one parameterization of the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\text{.}$$
2. Find a parameterization of the ellipse $$\frac{x^2}{4} + \frac{y^2}{16} = 1$$ that is traversed counterclockwise.
3. Find a parameterization of the ellipse $$\frac{(x+3)^2}{4} + \frac{(y-2)^2}{9} = 1\text{.}$$
4. Determine the $$x$$-$$y$$ equation of the ellipse that is parameterized by
$\mathbf{r}(t) = \langle 3 + 4\sin(2t), 1 + 3\cos(2t) \rangle. \nonumber$
13.

Consider the two-variable function $$z = f(x,y) = 3x^2 + 4y^2 - 2\text{.}$$

1. Determine a vector-valued function $$\mathbf{r}$$ that parameterizes the curve which is the $$x = 2$$ trace of $$z = f(x,y)\text{.}$$ Plot the resulting curve. Do likewise for $$x = -2, -1, 0,$$ and $$1\text{.}$$
2. Determine a vector-valued function $$\mathbf{r}$$ that parameterizes the curve which is the $$y = 2$$ trace of $$z = f(x,y)\text{.}$$ Plot the resulting curve. Do likewise for $$y = -2, -1, 0,$$ and $$1\text{.}$$
3. Determine a vector-valued function $$\mathbf{r}$$ that parameterizes the curve which is the $$z = 2$$ contour of $$z = f(x,y)\text{.}$$ Plot the resulting curve. Do likewise for $$z = -2, -1, 0,$$ and $$1\text{.}$$
4. Use the traces and contours you've just investigated to create a wireframe plot of the surface generated by $$z = f(x,y)\text{.}$$ In addition, write two sentences to describe the characteristics of the surface.
14.

Recall that any line in space may be represented parametrically by a vector-valued function.

1. Find a vector-valued function $$\mathbf{r}$$ that parameterizes the line through $$(-2,1,4)$$ in the direction of the vector $$\mathbf{v} = \langle 3, 2, -5 \rangle\text{.}$$
2. Find a vector-valued function $$\mathbf{r}$$ that parameterizes the line of intersection of the planes $$x + 2y - z = 4$$ and $$3x + y - 2z = 1\text{.}$$
3. Determine the point of intersection of the lines given by
$x = 2 + 3t, \ y = 1 - 2t, \ z = 4t, \nonumber$
$x = 3 + 1s, \ y = 3-2s, \ z = 2s. \nonumber$

Then, find a vector-valued function $$\mathbf{r}$$ that parameterizes the line that passes through the point of intersection you just found and is perpendicular to both of the given lines.

15.

For each of the following, describe the effect of the parameter $$s$$ on the parametric curve for $$t$$ in the interval $$[0,2 \pi]\text{.}$$

1. $$\displaystyle \mathbf{r}(t) = \langle \cos(t), \sin(t) + s \rangle$$
2. $$\displaystyle \mathbf{r}(t) = \langle \cos(t)-s, \sin(t) \rangle$$
3. $$\displaystyle \mathbf{r}(t) = \langle s\cos(t), \sin(t) \rangle$$
4. $$\displaystyle \mathbf{r}(t) = \langle s\cos(t), s\sin(t) \rangle$$
5. $$\displaystyle \mathbf{r}(t) = \langle \cos(st), \sin(st) \rangle$$

9.7: Derivatives and Integrals of Vector-Valued Functions

The WeBWorK problems are written by many different authors. Some authors use parentheses when writing vectors, e.g., $$(x(t),y(t),z(t))$$ instead of angle brackets $$\langle x(t),y(t),z(t) \rangle\text{.}$$ Please keep this in mind when working WeBWorK exercises.

1.

If $$\mathbf{r} (t) = \cos (3 t) \mathbf{i} + \sin (3 t) \mathbf{j} - 8 t \mathbf{k}\text{,}$$ compute:

A. The velocity vector $$\mathbf{v} (t) =$$ $$\mathbf{i} +$$ $$\mathbf{j} +$$ $$\mathbf{k}$$

B. The acceleration vector $$\mathbf{a} (t) =$$ $$\mathbf{i} +$$ $$\mathbf{j} +$$ $$\mathbf{k}$$

Note: the coefficients in your answers must be entered in the form of expressions in the variable $$t\text{;}$$ e.g. “5 cos(2t)”

$$-3\sin\!\left(3t\right)$$

$$3\cos\!\left(3t\right)$$

$$-8$$

$$-9\cos\!\left(3t\right)$$

$$-9\sin\!\left(3t\right)$$

$$0$$

2.

Given that the acceleration vector is $$\mathbf{a} \left( t \right) = \left( -9 \cos \left( 3 t \right) \right) \mathbf{i} + \left( -9 \sin \left( 3 t \right) \right) \mathbf{j} + \left( 3 t \right) \mathbf{k}\text{,}$$ the initial velocity is $$\mathbf{v} \left( 0 \right) = \mathbf{i + k}\text{,}$$ and the initial position vector is $$\mathbf{r} \left( 0 \right) = \mathbf{i + j + k}\text{,}$$ compute:

A. The velocity vector $$\mathbf{v} \left( t \right) =$$ $$\mathbf{i} +$$ $$\mathbf{j} +$$ $$\mathbf{k}$$

B. The position vector $$\mathbf{r} \left( t \right) =$$ $$\mathbf{i} +$$ $$\mathbf{j} +$$ $$\mathbf{k}$$

Note: the coefficients in your answers must be entered in the form of expressions in the variable \emph{t}; e.g. “5 cos(2t)”

$$-3\sin\!\left(3t\right)+1$$

$$3\cos\!\left(3t\right)-3$$

$$\frac{3t^{2}}{2}+1$$

$$\cos\!\left(3t\right)+t$$

$$\sin\!\left(3t\right)-3t+1$$

$$\frac{3t^{3}}{6}+t+1$$

3.

Evaluate

$$\int_{0}^{9}(t\mathbf{i}+ t^2\mathbf{j}+t^3\mathbf{k})dt$$ = $$\mathbf{i}+$$ $$\mathbf{j}+$$ $$\mathbf{k}\text{.}$$

$$40.5$$

$$243$$

$$1640.25$$

4.

Find parametric equations for line that is tangent to the curve $$x=\cos t,\ y=\sin t, \ z=t$$ at the point

$$(\cos(\frac{5\pi}{6}) ,\sin(\frac{5\pi}{6}) ,\frac{5\pi}{6} )$$.

Parametrize the line so that it passes through the given point at t=0. All three answers are required for credit.

$$x(t)$$ =

$$y(t)$$ =

$$z(t)$$ =

$$\cos\!\left(\frac{5\pi }{6}\right)+-\sin\!\left(\frac{5\pi }{6}\right)t;\,\sin\!\left(\frac{5\pi }{6}\right)+\cos\!\left(\frac{5\pi }{6}\right)t;\,\frac{5\pi }{6}+t$$

5.

If $$\mathbf{r}(t)= \cos(-5 t)\mathbf{i}+\sin( -5 t)\mathbf{j}+6 t \mathbf{k}$$

compute $$\mathbf{r}'(t)$$= $$\mathbf{i}+$$ $$\mathbf{j}+$$ $${\mathbf{k}}$$

and $$\int{\mathbf{r}}(t)\, dt$$= $$\mathbf{i}+$$ $$\mathbf{j}+$$ $$\mathbf{k} + \mathbf{C}$$

with $$\mathbf{C}$$ a constant vector.

$$5\sin\!\left(-5t\right)$$

$$-5\cos\!\left(-5t\right)$$

$$6$$

$$-0.2\sin\!\left(-5t\right)$$

$$0.2\cos\!\left(-5t\right)$$

$$3tt$$

6.

For the given position vectors $$\mathbf{r}(t)\text{,}$$

compute the (tangent) velocity vector $$\mathbf{r}'(t)$$ for the given value of $$t$$.

A) $$\displaystyle \textrm{Let } \mathbf{r}(t)= (\cos t,\, \sin t )\text{.}$$

Then $$\mathbf{r}'(\frac{\pi}{4})$$= ( , )?

B) $$\displaystyle \textrm{Let } {\mathbf{r}}(t)= (t^2,t^3)\text{.}$$

Then $${\mathbf{r}}'(2)$$= ( , )?

C) $$\displaystyle \textrm{Let } \mathbf{r}(t)= e^{t}\mathbf{i}+ e^{-2t}\mathbf{j}+ t\mathbf{k}\text{.}$$

Then $$\mathbf{r}'(1)$$= $$\mathbf{i}+$$ $$\mathbf{j}+$$ $$\mathbf{k}$$ ?

$$-0.707106781186547$$

$$0.707106781186548$$

$$4$$

$$12$$

$$2.71828182845905$$

$$-0.270670566473225$$

$$1$$

7.

Suppose $$\vec{r}(t) = \cos\!\left(\pi t\right)\boldsymbol{i}+\sin\!\left(\pi t\right)\boldsymbol{j}+3t\boldsymbol{k}$$ represents the position of a particle on a helix, where $$z$$ is the height of the particle.

(a) What is $$t$$ when the particle has height $$6\text{?}$$

$$t =$$

(b) What is the velocity of the particle when its height is $$6\text{?}$$

$$\vec{v} =$$

(c) When the particle has height $$6\text{,}$$ it leaves the helix and moves along the tangent line at the constant velocity found in part (b). Find a vector parametric equation for the position of the particle (in terms of the original parameter $$t$$) as it moves along this tangent line.

$$L(t) =$$

$$2$$

$$3.14159\boldsymbol{j}+3\boldsymbol{k}$$

$$\boldsymbol{i}+6\boldsymbol{k}+\left(t-2\right)\cdot 3.14159\boldsymbol{j}+3\boldsymbol{k}$$

8.

Suppose the displacement of a particle in motion at time $$t$$ is given by the parametric equations

$$x(t) = \left(3t-1\right)^{2}, \quad y(t) = 7, \quad z(t) = 54t^{3}-27t^{2}.$$

(a) Find the speed of the particle when $$t = 3\text{.}$$

Speed =

(b) Find $$t$$ when the particle is stationary.

$$t$$ =

$$1296.89$$

$$0.333333$$

9.

Find the derivative of the vector function

$$\mathbf r(t) = t\mathbf a \times (\mathbf b + t\mathbf c)\text{,}$$ where

$$\mathbf a = \langle -2, 4, 4\rangle\text{,}$$ $$\mathbf b = \langle 4, -4, -3\rangle\text{,}$$ and $$\mathbf c = \langle 3, -3, -5\rangle\text{.}$$

$$\mathbf r'(t) = \langle$$, , $$\rangle$$

$$4+2t\!\left(-8\right)$$

$$10+2t\cdot 2$$

$$-8+2t\!\left(-6\right)$$

10.

Let $$\mathbf{c}_1(t) = (e^{2t}, \sin(5t), -2t^3)\text{,}$$ and $$\mathbf{c}_2(t) = (e^{t}, \cos(4t), 2t^3)$$

$$\displaystyle \frac{d}{dt}\left[ \mathbf{c}_1(t) \cdot \mathbf{c}_2(t)\right] =$$

$$\displaystyle \frac{d}{dt}\left[ \mathbf{c}_1(t) \times \mathbf{c}_2(t)\right] =$$ $$\mathbf{i}\ +$$

$$\hspace{1.15in}$$ $$\mathbf{j}\ +$$

$$\hspace{1.15in}$$ $$\mathbf{k}$$

$$3\exp\!\left(3t\right)+5\cos\!\left(5t\right)\cos\!\left(4t\right)-4\sin\!\left(5t\right)\sin\!\left(4t\right)+-24t^{5}$$

$$6t^{2}\sin\!\left(5t\right)+10t^{3}\cos\!\left(5t\right)+-8t^{3}\sin\!\left(4t\right)--6t^{2}\cos\!\left(4t\right)$$

$$-6t^{2}\exp\!\left(1t\right)+-2t^{3}\exp\!\left(1t\right)-4t^{3}\exp\!\left(2t\right)-6t^{2}\exp\!\left(2t\right)$$

$$-4\exp\!\left(2t\right)\sin\!\left(4t\right)+2\exp\!\left(2t\right)\cos\!\left(4t\right)-5\exp\!\left(1t\right)\cos\!\left(5t\right)-1\exp\!\left(1t\right)\sin\!\left(5t\right)$$

11.

A gun has a muzzle speed of 90 meters per second. What angle of elevation should be used to hit an object 180 meters away? Neglect air resistance and use $$g = 9.8\, \textrm{m}/\textrm{sec}^{2}$$ as the acceleration of gravity.

12.

A child wanders slowly down a circular staircase from the top of a tower. With $$x,y,z$$ in feet and the origin at the base of the tower, her position $$t$$ minutes from the start is given by

$x=30\cos t,\quad y = 30 \sin t, \quad z = 80 - 5 t. \nonumber$

(a) How tall is the tower?

height = ft

(b) When does the child reach the bottom?

time = minutes

(c) What is her speed at time $$t\text{?}$$

speed = ft/min

(d) What is her acceleration at time $$t\text{?}$$

acceleration = ft/min$${}^2$$

$$80$$

$$\frac{80}{5}$$

$$30.4138$$

$$-30\cos\!\left(t\right)\,\mathit{\vec i}-30\sin\!\left(t\right)\,\mathit{\vec j}+0\,\mathit{\vec k}$$

13.

Compute the derivative of each of the following functions in two different ways: (1) use the rules provided in the theorem stated just after Activity 9.7.3, and (2) rewrite each given function so that it is stated as a single function (either a scalar function or a vector-valued function with three components), and differentiate component-wise. Compare your answers to ensure that they are the same.

1. $$\displaystyle \mathbf{r}(t) = \sin(t) \langle 2t, t^2, \arctan(t) \rangle$$
2. $$\mathbf{s}(t) = \mathbf{r}(2^t)\text{,}$$ where $$\mathbf{r}(t) = \langle t+2, \ln(t), 1 \rangle\text{.}$$
3. $$\displaystyle \mathbf{r}(t) = \langle \cos(t), \sin(t), t \rangle \cdot \langle -\sin(t), \cos(t), 1 \rangle$$
4. $$\displaystyle \mathbf{r}(t) = \langle \cos(t), \sin(t), t \rangle \times \langle -\sin(t), \cos(t), 1 \rangle$$
14.

Consider the two vector-valued functions given by

$\mathbf{r}(t) = \left\langle t + 1, \cos\left(\frac{\pi}{2} t\right), \frac{1}{1+t} \right\rangle \nonumber$

and

$\mathbf{w}(s) = \left\langle s^2, \sin\left(\frac{\pi}{2}s\right), s \right\rangle. \nonumber$
1. Determine the point of intersection of the curves generated by $$\mathbf{r}(t)$$ and $$\mathbf{w}(s)\text{.}$$ To do so, you will have to find values of $$a$$ and $$b$$ that result in $$\mathbf{r}(a)$$ and $$\mathbf{w}(b)$$ being the same vector.
2. Use the value of $$a$$ you determined in (a) to find a vector form of the tangent line to $$\mathbf{r}(t)$$ at the point where $$t = a\text{.}$$
3. Use the value of $$b$$ you determined in (a) to find a vector form of the tangent line to $$\mathbf{w}(s)$$ at the point where $$s = b\text{.}$$
4. Suppose that $$z = f(x,y)$$ is a function that generates a surface in three-dimensional space, and that the curves generated by $$\mathbf{r}(t)$$ and $$\mathbf{w}(s)$$ both lie on this surface. Note particularly that the point of intersection you found in (a) lies on this surface. In addition, observe that the two tangent lines found in (b) and (c) both lie in the tangent plane to the surface at the point of intersection. Use your preceding work to determine the equation of this tangent plane.
15.

In this exercise, we determine the equation of a plane tangent to the surface defined by $$f(x,y) = \sqrt{x^2+y^2}$$ at the point $$(3,4,5)\text{.}$$

1. Find a parameterization for the $$x=3$$ trace of $$f\text{.}$$ What is a direction vector for the line tangent to this trace at the point $$(3,4,5)\text{?}$$
2. Find a parameterization for the $$y=4$$ trace of $$f\text{.}$$ What is a direction vector for the line tangent to this trace at the point $$(3,4,5)\text{?}$$
3. The direction vectors in parts (a) and (b) form a plane containing the point $$(3,4,5)\text{.}$$ What is a normal vector for this plane?
4. Use your work in parts (a), (b), and (c) to determine an equation for the tangent plane. Then, use appropriate technology to draw the graph of $$f$$ and the plane you determined on the same set of axes. What do you observe? (We will discuss tangent planes in more detail in Chapter 10.)
16.

For each given function $$\mathbf{r}\text{,}$$ determine $$\int \mathbf{r}(t) \ dt\text{.}$$ In addition, recalling the Fundamental Theorem of Calculus for functions of a single variable, also evaluate $$\int_0^1 \mathbf{r}(t) \ dt$$ for each given function $$\mathbf{r}\text{.}$$ Is the resulting quantity a scalar or a vector? What does it measure?

1. $$\displaystyle \mathbf{r}(t) = \left\langle \cos(t), \frac{1}{t+1}, te^t \right\rangle$$
2. $$\displaystyle \mathbf{r}(t) = \left\langle \cos(3t), \sin(2t), t \right\rangle$$
3. $$\displaystyle \mathbf{r}(t) = \left\langle \frac{t}{1+t^2}, te^{t^2}, \frac{1}{1+t^2} \right\rangle$$
17.

In this exercise, we develop the formula for the position function of a projectile that has been launched at an initial speed of $$|\mathbf{v}_0|$$ and a launch angle of $$\theta.$$ Recall that $$\mathbf{a}(t) = \langle 0, -g \rangle$$ is the constant acceleration of the projectile at any time $$t\text{.}$$

1. Find all velocity vectors for the given acceleration vector $$\mathbf{a}\text{.}$$ When you anti-differentiate, remember that there is an arbitrary constant that arises in each component.
2. Use the given information about initial speed and launch angle to find $$\mathbf{v}_0\text{,}$$ the initial velocity of the projectile. You will want to write the vector in terms of its components, which will involve $$\sin(\theta)$$ and $$\cos(\theta)\text{.}$$
3. Next, find the specific velocity vector function $$\mathbf{v}$$ for the projectile. That is, combine your work in (a) and (b) in order to determine expressions in terms of $$|\mathbf{v}_0|$$ and $$\theta$$ for the constants that arose when integrating.
4. Find all possible position vectors for the velocity vector $$\mathbf{v}(t)$$ you determined in (c).
5. Let $$\mathbf{r}(t)$$ denote the position vector function for the given projectile. Use the fact that the object is fired from the position $$(x_0, y_0)$$ to show it follows that
$\mathbf{r}(t) = \left\langle |\mathbf{v}_0| \cos(\theta)t + x_0, -\frac{g}{2}t^2 + |\mathbf{v}_0| \sin(\theta)t + y_0 \right\rangle. \nonumber$
18.

A central force is one that acts on an object so that the force $$\mathbf{F}$$ is parallel to the object's position $$\mathbf{r}\text{.}$$ Since Newton's Second Law says that an object's acceleration is proportional to the force exerted on it, the acceleration $$\mathbf{a}$$ of an object moving under a central force will be parallel to its position $$\mathbf{r}\text{.}$$ For instance, the Earth's acceleration due to the gravitational force that the sun exerts on the Earth is parallel to the Earth's position vector as shown in Figure 9.7.8.

1. If an object of mass $$m$$ is moving under a central force, the angular momentum vector is defined to be $$\mathbf{L}=m\mathbf{r}\times\mathbf{v}\text{.}$$ Assuming the mass is constant, show that the angular momentum is constant by showing that
$\frac{d\mathbf{L}}{dt} = \mathbf{0}. \nonumber$
2. Explain why $$\mathbf{L}\cdot\mathbf{r} = 0\text{.}$$
3. Explain why we may conclude that the object is constrained to lie in the plane passing through the origin and perpendicular to $$\mathbf{L}\text{.}$$

9.8: Arc Length and Curvature

The WeBWorK problems are written by many different authors. Some authors use parentheses when writing vectors, e.g., $$(x(t),y(t),z(t))$$ instead of angle brackets $$\langle x(t),y(t),z(t) \rangle\text{.}$$ Please keep this in mind when working WeBWorK exercises.

1.

Find the length of the curve

$x = 5+5t,\quad y = 3+4t, z = t-4, \nonumber$

for $$2 \le t \le 3\text{.}$$

length =

(Think of second way that you could calculate this length, too, and see that you get the same result.)

$$\sqrt{5\cdot 5+4\cdot 4+1\cdot 1}\!\left(3-2\right)$$

2.

Consider the curve $$\displaystyle \mathbf{r} = (e^{-5 t} \cos(2 t), e^{-5 t} \sin(2 t), e^{-5 t})\text{.}$$

Compute the arclength function $$s(t)\text{:}$$ (with initial point $$t=0$$).

$$\frac{7.34847\!\left(\exp\!\left(-5t\right)-1\right)}{-5}$$

3.

Find the length of the given curve:

$\mathbf{r} \left( t \right) = \left( -3 t, -3 \sin t, -3 \cos t \right) \nonumber$

where $$-4 \leq t \leq 1\text{.}$$

$$21.2132034355964$$

4.

Find the curvature of $$y=\sin \left( -2 x \right)$$ at $$x = \frac{\pi}{4}\text{.}$$

$$4$$

5.

Consider the path $$\mathbf{r}(t) = (10 t, 5 t^2, 5\ln t)$$ defined for $$t > 0\text{.}$$

Find the length of the curve between the points $$(10, 5, 0)$$ and $$(40, 80, 5 \ln(4))\text{.}$$

$$75+5\ln\!\left(4\right)$$

6.

Find the curvature $$\kappa (t)$$ of the curve $$\mathbf{r} (t) = \left( -3 \sin t \right) \mathbf{i} + \left( -3 \sin t \right) \mathbf{j} + \left( 1 \cos t \right) \mathbf{k}$$

$$\frac{4.24264}{\left(\left(1\sin\!\left(t\right)\right)^{2}+2\!\left(-3\cos\!\left(t\right)\right)^{2}\right)^{1.5}}$$

7.

A factory has a machine which bends wire at a rate of 9 unit(s) of curvature per second. How long does it take to bend a straight wire into a circle of radius 7?

seconds

$$0.0158730158730159$$

8.

Find the unit tangent vector at the indicated point of the vector function

$\mathbf{r}(t) = e^{16 t}\cos t \,\mathbf i + e^{16 t}\sin t \,\mathbf j + e^{16 t} \,\mathbf k \nonumber$

$$\mathbf T(\pi/2) = \langle$$, , $$\rangle$$

$$-0.0441510785688348$$

$$0.706417257101357$$

$$0.706417257101357$$

9.

Consider the vector function

$$\mathbf r(t) = \langle t, t^{8}, t^{3}\rangle$$

Compute

$$\mathbf r'(t) = \langle$$, , $$\rangle$$

$$\mathbf T(1) = \langle$$, , $$\rangle$$

$$\mathbf r''(t) = \langle$$, , $$\rangle$$

$$\mathbf r'(t)\times \mathbf r''(t) = \langle$$, , $$\rangle$$

$$1$$

$$8t^{7}$$

$$3t^{2}$$

$$0.116247638743819$$

$$0.929981109950554$$

$$0.348742916231458$$

$$0$$

$$56t^{6}$$

$$6t^{1}$$

$$-120t^{8}$$

$$-6t^{1}$$

$$56t^{6}$$

10.

Starting from the point $$\left( 5, 4, -2 \right)\text{,}$$ reparametrize the curve

$$\mathbf{x} (t) = ( 5 - 2t, 4 + 2t, -2 + t )$$ in terms of arclength.

$$\mathbf{y}(s) = ($$, , $$)$$

$$5+\frac{s\!\left(-2\right)}{3}$$

$$4+\frac{s\cdot 2}{3}$$

$$-2+\frac{s\cdot 1}{3}$$

11.

Consider the moving particle whose position at time $$t$$ in seconds is given by the vector-valued function $$\mathbf{r}$$ defined by $$\mathbf{r}(t) = 5t \mathbf{i} + 4\sin(3t) \mathbf{j} + 4\cos(3t) \mathbf{k}\text{.}$$ Use this function to answer each of the following questions.

1. Find the unit tangent vector, $$\mathbf{T}(t)\text{,}$$ to the space curve traced by $$\mathbf{r}(t)$$ at time $$t\text{.}$$ Write one sentence that explains what $$\mathbf{T}(t)$$ tells us about the particle's motion.
2. Determine the speed of the particle moving along the space curve with the given parameterization.
3. Find the exact distance traveled by the particle on the time interval $$[0,\pi/3]\text{.}$$
4. Find the average velocity of the particle on the time interval $$[0, \pi/3]\text{.}$$
5. Determine the parameterization of the given curve with respect to arc length.
12.

Let $$y = f(x)$$ define a curve in the plane. We can consider this curve as a curve in three-space with $$z$$-coordinate 0.

1. Find a parameterization of the form $$\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle$$ of the curve $$y=f(x)$$ in three-space.
2. Use the formula
$\kappa = \frac{\lvert \mathbf{r}'(t) \times \mathbf{r}''(t) \rvert}{\lvert \mathbf{r}'(t) \rvert^3} \nonumber$

to show that

$\kappa = \frac{\lvert f''(x) \rvert}{\left[1+(f'(x))^2\right]^{3/2}}. \nonumber$
13.

Consider the single variable function defined by $$y = 4x^2 - x^3.$$

1. Find a parameterization of the form $$\mathbf{r}(t) = \langle x(t), y(t) \rangle$$ that traces the curve $$y = 4x^2 - x^3$$ on the interval from $$x = -3$$ to $$x = 3\text{.}$$
2. Write a definite integral which, if evaluated, gives the exact length of the given curve from $$x = -3$$ to $$x = 3\text{.}$$ Why is the integral difficult to evaluate exactly?
3. Determine the curvature, $$\kappa(t)\text{,}$$ of the parameterized curve. (Exercise 9.8.5.12 might be useful here.)
4. Use appropriate technology to approximate the absolute maximum and minimum of $$\kappa(t)$$ on the parameter interval for your parameterization. Compare your results with the graph of $$y = 4x^2 - x^3\text{.}$$ How do the absolute maximum and absolute minimum of $$\kappa(t)$$ align with the original curve?
14.

Consider the standard helix parameterized by $$\mathbf{r}(t) = \cos(t) \mathbf{i} + \sin(t) \mathbf{j} + t \mathbf{k}\text{.}$$

1. Recall that the unit tangent vector, $$\mathbf{T}(t)\text{,}$$ is the vector tangent to the curve at time $$t$$ that points in the direction of motion and has length 1. Find $$\mathbf{T}(t)\text{.}$$
2. Explain why the fact that $$| \mathbf{T}(t) | = 1$$ implies that $$\mathbf{T}$$ and $$\mathbf{T}'$$ are orthogonal vectors for every value of $$t\text{.}$$ (Hint: note that $$\mathbf{T} \cdot \mathbf{T} = |\mathbf{T}|^2 = 1,$$ and compute $$\frac{d}{dt}[\mathbf{T} \cdot \mathbf{T}]\text{.}$$)
3. For the given function $$\mathbf{r}$$ with unit tangent vector $$\mathbf{T}(t)$$ (from (a)), determine $$\mathbf{N}(t) = \frac{1}{|\mathbf{T}'(t)|} \mathbf{T}'(t)\text{.}$$
4. What geometric properties does $$\mathbf{N}(t)$$ have? That is, how long is this vector, and how is it situated in comparison to $$\mathbf{T}(t)\text{?}$$
5. Let $$\mathbf{B}(t) = \mathbf{T}(t) \times \mathbf{N}(t)\text{,}$$ and compute $$\mathbf{B}(t)$$ in terms of your results in (a) and (c).
6. What geometric properties does $$\mathbf{B}(t)$$ have? That is, how long is this vector, and how is it situated in comparison to $$\mathbf{T}(t)$$ and $$\mathbf{N}(t)\text{?}$$
7. Sketch a plot of the given helix, and compute and sketch $$\mathbf{T}(\pi/2)\text{,}$$ $$\mathbf{N}(\pi/2)\text{,}$$ and $$\mathbf{B}(\pi/2)\text{.}$$
15.

In this exercise we verify the curvature formula

$\kappa = \frac{\lvert \mathbf{r}'(t) \times \mathbf{r}''(t) \rvert}{\lvert \mathbf{r}'(t) \rvert^3}. \nonumber$
1. Explain why
$\lvert \mathbf{r}'(t) \rvert = \frac{ds}{dt}. \nonumber$
2. Use the fact that $$\mathbf{T}(t) = \frac{\mathbf{r}'(t)}{\lvert \mathbf{r}'(t) \rvert}$$ and $$\lvert \mathbf{r}'(t) \rvert = \frac{ds}{dt}$$ to explain why
$\mathbf{r}'(t) = \frac{ds}{dt} \mathbf{T}(t). \nonumber$
3. The Product Rule shows that
$\mathbf{r}''(t) = \frac{d^2s}{dt^2} \mathbf{T}(t) + \frac{ds}{dt} \mathbf{T}'(t). \nonumber$

Explain why

$\mathbf{r}'(t) \times \mathbf{r}''(t) = \left(\frac{ds}{dt}\right)^2 (\mathbf{T}(t) \times \mathbf{T}'(t)). \nonumber$
4. In Exercise 9.8.5.14 we showed that $$\lvert \mathbf{T}(t) \rvert = 1$$ implies that $$\mathbf{T}(t)$$ is orthogonal to $$\mathbf{T}'(t)$$ for every value of $$t\text{.}$$ Explain what this tells us about $$\lvert \mathbf{T}(t) \times \mathbf{T}'(t) \rvert$$ and conclude that
$\lvert \mathbf{r}'(t) \times \mathbf{r}''(t) \rvert = \left(\frac{ds}{dt}\right)^2 \lvert \mathbf{T}'(t) \rvert. \nonumber$
5. Finally, use the fact that $$\kappa = \frac{\lvert \mathbf{T}'(t) \rvert }{\lvert \mathbf{r}'(t) \rvert}$$ to verify that
$\kappa = \frac{\lvert \mathbf{r}'(t) \times \mathbf{r}''(t) \rvert}{\lvert \mathbf{r}'(t) \rvert^3}. \nonumber$
16.

In this exercise we explore how to find the osculating circle for a given curve. As an example, we will use the curve defined by $$f(x) = x^2\text{.}$$ Recall that this curve can be parameterized by $$x(t) = t$$ and $$y(t)=t^2\text{.}$$

1. Use (9.8.5) to find $$\mathbf{T}(t)$$ for our function $$f\text{.}$$
2. To find the center of the osculating circle, we will want to find a vector that points from a point on the curve to the center of the circle. Such a vector will be orthogonal to the tangent vector at that point. Recall that $$\mathbf{T}(s) = \langle \cos(\phi(s)), \sin(\phi(s)) \rangle\text{,}$$ where $$\phi$$ is the angle the tangent vector to the curve makes with a horizontal vector. Use this fact to show that
$\mathbf{T} \cdot \frac{dT}{ds} = 0. \nonumber$

Explain why this tells us that $$\frac{dT}{ds}$$ is orthogonal to $$\mathbf{T}\text{.}$$ Let $$\mathbf{N}$$ be the unit vector in the direction of $$\frac{dT}{ds}\text{.}$$ The vector $$\mathbf{N}$$ is called the principal unit normal vector and points in the direction toward which the curve is turning. The vector $$\mathbf{N}$$ also points toward the center of the osculating circle.

3. Find $$\mathbf{T}$$ at the point $$(1,1)$$ on the graph of $$f\text{.}$$ Then find $$\mathbf{N}$$ at this same point. How do you know you have the correct direction for $$\mathbf{N}\text{?}$$
4. Let $$P$$ be a point on the curve. Recall that $$\rho = \frac{1}{\kappa}$$ at point $$P$$ is the radius of the osculating circle at point $$P\text{.}$$ We call $$\rho$$ the radius of curvature at point $$P\text{.}$$ Let $$C$$ be the center of the osculating circle to the curve at point $$P\text{,}$$ and let $$O$$ be the origin. Let $$\mathbf{\gamma}$$ be the vector $$\overrightarrow{OC}\text{.}$$ See Figure 9.8.8 for an illustration using an arbitrary function $$f\text{.}$$

Which vector, in terms of $$\rho$$ and $$\mathbf{N}$$ points from the point $$P$$ to the point $$C\text{?}$$ Use this vector to explain why
$\mathbf{\gamma} = \mathbf{r} + \rho \mathbf{N}, \nonumber$

where $$\mathbf{r} = \overrightarrow{OP}\text{.}$$

5. Finally, use the previous work to find the center of the osculating circle for $$f$$ at the point $$(1,1)\text{.}$$ Draw pictures of the curve and the osculating circle to verify your work.

9: Multivariable and Vector Functions is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.