5.4E: Excersies

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Exercise $\PageIndex{1}$

For the following exercises, the vectors $\displaystyle u$ and $\displaystyle v$ are given.

a. Find the cross product $\displaystyle u×v$ of the vectors $\displaystyle u$ and $\displaystyle v$. Express the answer in component form.

b. Sketch the vectors $\displaystyle u,v,$ and $\displaystyle u×v.$

1) $\displaystyle u=⟨2,0,0⟩, v=⟨2,2,0⟩$

Answer

$\displaystyle a. u×v=⟨0,0,4⟩;$

$This figure is the first octant of the 3-dimensional coordinate system. On the x-axis there is a vector labeled “u.” In the x y-plane there is a vector labeled “v.” On the z-axis there is the vector labeled “u cross v.”$

2) $\displaystyle u=⟨3,2,−1⟩, v=⟨1,1,0⟩$

3) $\displaystyle u=2i+3j, v=j+2k$

Answer

a. $\displaystyle a. u×v=⟨6,−4,2⟩;$

$This figure is the first octant of the 3-dimensional coordinate system and shows three vectors. The first vector is labeled u and has components <2, 3, 0>. The second vector is labeled v and has components <0, 1, 2>.” The third vector is labeled u cross v and has components <6, -4, 2>.”$

4) $\displaystyle u=2j+3k, v=3i+k$

5) Simplify $\displaystyle (i×i−2i×j−4i×k+3j×k)×i.$

Answer: $\displaystyle −2j−4k$

6) Simplify $\displaystyle j×(k×j+2j×i−3j×j+5i×k).$

Exercise $\PageIndex{2}$

In the following exercises, vectors $\displaystyle u$ and $\displaystyle v$ are given. Find unit vector $\displaystyle w$ in the direction of the cross product vector $\displaystyle u×v.$ Express your answer using standard unit vectors.

7) $\displaystyle u=⟨3,−1,2⟩, v=⟨−2,0,1⟩$

Answer: $\displaystyle w=−\frac{1}{3\sqrt{6}}i−\frac{7}{3\sqrt{6}}j−\frac{2}{3\sqrt{6}}k$

8) $\displaystyle u=⟨2,6,1⟩, v=⟨3,0,1⟩$

9) $\displaystyle u=\vec{AB}, v=\vec{AC},$ where $\displaystyle A(1,0,1), B(1,−1,3)$, and $\displaystyle C(0,0,5)$

Answer: $\displaystyle w=−\frac{4}{\sqrt{21}}i−\frac{2}{\sqrt{21}}j−\frac{1}{\sqrt{21}}k$

10) $\displaystyle u=\vec{OP}, v=\vec{PQ},$ where $\displaystyle P(−1,1,0)$ and $\displaystyle Q(0,2,1)$

Exercise $\PageIndex{3}$

11) Determine the real number $\displaystyle α$ such that $\displaystyle u×v$ and $\displaystyle i$ are orthogonal, where $\displaystyle u=3i+j−5k$ and $\displaystyle v=4i−2j+αk.$

Answer: $\displaystyle α=10$

12) Show that $\displaystyle u×v$ and $\displaystyle 2i−14j+2k$ cannot be orthogonal for any α real number, where $\displaystyle u=i+7j−k$ and $\displaystyle v=αi+5j+k$.

13) Show that $\displaystyle u×v$ is orthogonal to $\displaystyle u+v$ and $\displaystyle u−v$, where $\displaystyle u$ and $\displaystyle v$ are nonzero vectors.

14) Show that $\displaystyle v×u$ is orthogonal to $\displaystyle (u⋅v)(u+v)+u$, where $\displaystyle u$ and $\displaystyle v$ are nonzero vectors.

15) Calculate the determinant $\displaystyle \begin{bmatrix}i&j&K\\1&&−1&7\\2&0&3\end{bmatrix}$.

Answer: $\displaystyle −3i+11j+2k$

16) Calculate the determinant $\displaystyle \begin{bmatrix}i&j&K\\0&3&−4\\1&6&−1\end{bmatrix}$.

Exercise $\PageIndex{4}$

For the following exercises, the vectors $\displaystyle u$ and $\displaystyle v$ are given. Use determinant notation to find vector $\displaystyle w$ orthogonal to vectors $\displaystyle u$ and $\displaystyle v$.

17) $\displaystyle u=⟨−1,0,e^t⟩, v=⟨1,e^{−t},0⟩,$ where $\displaystyle t$ is a real number

Answer: $\displaystyle w=⟨−1,e^t,−e^{−t}⟩$

18) $\displaystyle u=⟨1,0,x⟩, v=⟨\frac{2}{x},1,0⟩,$ where $\displaystyle x$ is a nonzero real number

19) Find vector $\displaystyle (a−2b)×c,$ where $\displaystyle a=\begin{bmatrix}i&j&k\\2&−1&5\\0&1&8\end{bmatrix}, b=\begin{bmatrix}i&j&K\\0&1&1\\2&−1&−2\end{bmatrix},$ and $\displaystyle c=i+j+k.$

Answer: $\displaystyle −26i+17j+9k$

20) Find vector $\displaystyle c×(a+3b),$ where $\displaystyle a=\begin{bmatrix}i&j&K\\5&0&9\\0&1&0\end{bmatrix}, b=\begin{bmatrix}i&j&k\\0&−1&1\\7&1&−1\end{bmatrix},$ and $\displaystyle c=i−k.$

Exercise $\PageIndex{5}$

21) [T] Use the cross product $\displaystyle u×v$ to find the acute angle between vectors $\displaystyle u$ and $\displaystyle v$, where $\displaystyle u=i+2j$ and $\displaystyle v=i+k.$ Express the answer in degrees rounded to the nearest integer.

Answer: $\displaystyle 72°$

22) [T] Use the cross product $\displaystyle u×v$ to find the obtuse angle between vectors $\displaystyle u$ and $\displaystyle v$, where $\displaystyle u=−i+3j+k$ and $\displaystyle v=i−2j.$ Express the answer in degrees rounded to the nearest integer.

Exercise $\PageIndex{6}$

23) Use the sine and cosine of the angle between two nonzero vectors $\displaystyle u$ and $\displaystyle v$ to prove Lagrange’s identity: $\displaystyle ‖u×v‖^2=‖u‖^2‖v‖^2−(u⋅v)^2$.

24) Verify Lagrange’s identity $\displaystyle ‖u×v‖^2=‖u‖^2‖v‖^2−(u⋅v)^2$ for vectors $\displaystyle u=−i+j−2k$ and $\displaystyle v=2i−j.$

25) Nonzero vectors $\displaystyle u$ and $\displaystyle v$ are called collinear if there exists a nonzero scalar $\displaystyle α$ such that $\displaystyle v=αu$. Show that $\displaystyle u$ and $\displaystyle v$ are collinear if and only if $\displaystyle u×v=0.$

26) Nonzero vectors $\displaystyle u$ and $\displaystyle v$ are called collinear if there exists a nonzero scalar $\displaystyle α$ such that $\displaystyle v=αu$. Show that vectors $\displaystyle \vec{AB}$ and $\displaystyle \vec{AC}$ are collinear, where $\displaystyle A(4,1,0), B(6,5,−2),$ and $\displaystyle C(5,3,−1).$

27) Find the area of the parallelogram with adjacent sides $\displaystyle u=⟨3,2,0⟩$ and $\displaystyle v=⟨0,2,1⟩$.

Answer: $\displaystyle 7$

28) Find the area of the parallelogram with adjacent sides $\displaystyle u=i+j$ and $\displaystyle v=i+k.$

Exercise $\PageIndex{7}$

29) Consider points $\displaystyle A(3,−1,2),B(2,1,5),$ and $\displaystyle C(1,−2,−2).$

a. Find the area of parallelogram ABCD with adjacent sides $\displaystyle \vec{AB}$ and $\displaystyle \vec{AC}$.

b. Find the area of triangle ABC.

c. Find the distance from point A to line BC.

Answer: $\displaystyle a. 5\sqrt{6}; b. \frac{5\sqrt{6}}{2}; c. \frac{5\sqrt{6}}{\sqrt{59}}$

30) Consider points $\displaystyle A(2,−3,4),B(0,1,2),$ and $\displaystyle C(−1,2,0).$

a. Find the area of parallelogram ABCD with adjacent sides $\displaystyle \vec{AB}$ and $\displaystyle \vec{AC}$.

b. Find the area of triangle ABC.

c. Find the distance from point B to line AC.

Exercise $\PageIndex{8}$

In the following exercises, vectors $\displaystyle u,v$, and $\displaystyle w$ are given.

a. Find the triple scalar product $\displaystyle u⋅(v×w).$

b. Find the volume of the parallelepiped with the adjacent edges $\displaystyle u,v$, and $\displaystyle w$.

31) $\displaystyle u=i+j, v=j+k,$ and $\displaystyle w=i+k$

Answer: $\displaystyle a. 2; b. 2$

32) $\displaystyle u=⟨−3,5,−1⟩, v=⟨0,2,−2⟩,$ and $\displaystyle w=⟨3,1,1⟩$

33) Calculate the triple scalar products $\displaystyle v⋅(u×w)$ and $\displaystyle w⋅(u×v),$ where $\displaystyle u=⟨1,1,1⟩, v=⟨7,6,9⟩,$ and $\displaystyle w=⟨4,2,7⟩.$

Answer: $\displaystyle v⋅(u×w)=−1, w⋅(u×v)=1$

34) Calculate the triple scalar products $\displaystyle w⋅(v×u)$ and $\displaystyle u⋅(w×v),$ where $\displaystyle u=⟨4,2,−1⟩, v=⟨2,5,−3⟩,$ and $\displaystyle w=⟨9,5,−10⟩.$

35) Find vectors $\displaystyle a,b$, and $\displaystyle c$ with a triple scalar product given by the determinant $\displaystyle \begin{bmatrix}1&2&3\\0&2&5\\8&9&2\end{bmatrix}$. Determine their triple scalar product.

Answer: $\displaystyle a=⟨1,2,3⟩, b=⟨0,2,5⟩, c=⟨8,9,2⟩; a⋅(b×c)=−9$

36) The triple scalar product of vectors $\displaystyle a,b,$ and $\displaystyle c$ is given by the determinant $\displaystyle \begin{bmatrix}0&−2&1\\0&1&4\\1&−3&7\end{bmatrix}$. Find vector $\displaystyle a−b+c.$

Exercise $\PageIndex{9}$

37) Consider the parallelepiped with edges $\displaystyle OA,OB,$ and $\displaystyle OC$, where $\displaystyle A(2,1,0),B(1,2,0),$ and $\displaystyle C(0,1,α).$

a. Find the real number $\displaystyle α>0$ such that the volume of the parallelepiped is $\displaystyle 3$ units³^.

b. For $\displaystyle α=1,$ find the height $\displaystyle h$ from vertex $\displaystyle C$ of the parallelepiped. Sketch the parallelepiped.

Answer

$\displaystyle a. α=1; b. h=1,$

$This figure is the first octant of the 3-dimensional coordinate system. There is a parallelepided drawn. From the origin there are three vectors to vertices on the parallelepiped. They are vectors to the points A (2, 1, 0); B (1, 2, 0); and C (0, 1, alpha).$

38) Consider points $\displaystyle A(α,0,0),B(0,β,0),$ and $\displaystyle C(0,0,γ)$, with $\displaystyle α, β$, and $\displaystyle γ$ positive real numbers.

a. Determine the volume of the parallelepiped with adjacent sides $\displaystyle \vec{OA}, \vec{OB},$ and $\displaystyle \vec{OC}$.

b. Find the volume of the tetrahedron with vertices $\displaystyle O,A,B,$ and $\displaystyle C$. (Hint: The volume of the tetrahedron is $\displaystyle 1/6$ of the volume of the parallelepiped.)

c. Find the distance from the origin to the plane determined by $\displaystyle A,B,$ and $\displaystyle C$. Sketch the parallelepiped and tetrahedron.

39) Let $\displaystyle u,v,$ and $\displaystyle w$ be three-dimensional vectors and c be a real number. Prove the following properties of the cross product.

a. $\displaystyle u×u=0$

b. $\displaystyle u×(v+w)=(u×v)+(u×w)$

c. $\displaystyle c(u×v)=(cu)×v=u×(cv)$

d. $\displaystyle u⋅(u×v)=0$

40) Show that vectors $\displaystyle u=⟨1,0,−8⟩, v=⟨0,1,6⟩,$ and $\displaystyle w=⟨−1,9,3⟩$ satisfy the following properties of the cross product.

a. $\displaystyle u×u=0$

b. $\displaystyle u×(v+w)=(u×v)+(u×w)$

c. $\displaystyle c(u×v)=(cu)×v=u×(cv)$

d. $\displaystyle u⋅(u×v)=0$

41) Nonzero vectors $\displaystyle u,v,$ and $\displaystyle w$ are said to be linearly dependent if one of the vectors is a linear combination of the other two. For instance, there exist two nonzero real numbers $\displaystyle α$ and $\displaystyle β$ such that $\displaystyle w=αu+βv$. Otherwise, the vectors are called linearly independent. Show that $\displaystyle u,v,$ and $\displaystyle w$ are coplanar if and only if they are linear dependent.

42) Consider vectors $\displaystyle u=⟨1,4,−7⟩, v=⟨2,−1,4⟩, w=⟨0,−9,18⟩,$ and $\displaystyle p=⟨0,−9,17⟩.$

a. Show that $\displaystyle u,v,$ and $\displaystyle w$ are coplanar by using their triple scalar product

b. Show that $\displaystyle u,v,$ and $\displaystyle w$ are coplanar, using the definition that there exist two nonzero real numbers $\displaystyle α$ and $\displaystyle β$ such that $\displaystyle w=αu+βv.$

c. Show that $\displaystyle u,v,$ and $\displaystyle p$ are linearly independent—that is, none of the vectors is a linear combination of the other two.

43) Consider points $\displaystyle A(0,0,2), B(1,0,2), C(1,1,2),$ and $\displaystyle D(0,1,2).$ Are vectors $\displaystyle \vec{AB}, \vec{AC},$ and $\displaystyle \vec{AD}$ linearly dependent (that is, one of the vectors is a linear combination of the other two)?

Answer: Yes, $\displaystyle \vec{AD}=α\vec{AB}+β\vec{AC},$ where $\displaystyle α=−1$ and $\displaystyle β=1.$

44) Show that vectors $\displaystyle i+j, i−j,$ and $\displaystyle i+j+k$ are linearly independent—that is, there exist two nonzero real numbers $\displaystyle α$ and $\displaystyle β$ such that $\displaystyle i+j+k=α(i+j)+β(i−j).$

45) Let $\displaystyle u=⟨u_1,u_2⟩$ and $\displaystyle v=⟨v_1,v_2⟩$ be two-dimensional vectors. The cross product of vectors $\displaystyle u$ and $\displaystyle v$ is not defined. However, if the vectors are regarded as the three-dimensional vectors $\displaystyle \tilde{u}=⟨u_1,u_2,0⟩$ and $\displaystyle \tilde{v}=⟨v_1,v_2,0⟩$, respectively, then, in this case, we can define the cross product of $\displaystyle \tilde{u}$ and $\displaystyle \tilde{v}$. In particular, in determinant notation, the cross product of $\displaystyle \tilde{u}$ and $\displaystyle \tilde{v}$ is given by

$\displaystyle \tilde{u}×\tilde{v}=\begin{bmatrix}i&j&k\\u_1&u_2&0\\v_1&v_2&0\end{bmatrix}$.

Use this result to compute $\displaystyle (icosθ+jsinθ)×(isinθ−jcosθ),$ where $\displaystyle θ$ is a real number.

Answer: $\displaystyle −k$

46) Consider points $\displaystyle P(2,1), Q(4,2),$ and $\displaystyle R(1,2).$

a. Find the area of triangle $\displaystyle P,Q,$ and $\displaystyle R.$

b. Determine the distance from point $\displaystyle R$ to the line passing through $\displaystyle P$ and $\displaystyle Q$.

47) Determine a vector of magnitude $\displaystyle 10$ perpendicular to the plane passing through the x-axis and point $\displaystyle P(1,2,4).$

Answer: $\displaystyle ⟨0,±4\sqrt{5},2\sqrt{5}⟩$

48) Determine a unit vector perpendicular to the plane passing through the z-axis and point $\displaystyle A(3,1,−2).$

49) Consider $\displaystyle u$ and $\displaystyle v$ two three-dimensional vectors. If the magnitude of the cross product vector $\displaystyle u×v$ is $\displaystyle k$ times larger than the magnitude of vector $\displaystyle u$, show that the magnitude of $\displaystyle v$ is greater than or equal to $\displaystyle k$, where $\displaystyle k$ is a natural number.

Exercise $\PageIndex{10}$

50) [T] Assume that the magnitudes of two nonzero vectors $\displaystyle u$ and $\displaystyle v$ are known. The function $\displaystyle f(θ)=‖u‖‖v‖sinθ$ defines the magnitude of the cross product vector $\displaystyle u×v,$ where $\displaystyle θ∈[0,π]$ is the angle between $\displaystyle u$ and $\displaystyle v$.

a. Graph the function $\displaystyle f$.

b. Find the absolute minimum and maximum of function $\displaystyle f$. Interpret the results.

c. If $\displaystyle ‖u‖=5$ and $\displaystyle ‖v‖=2$, find the angle between $\displaystyle u$ and $\displaystyle v$ if the magnitude of their cross product vector is equal to $\displaystyle 9$.

51) Find all vectors $\displaystyle w=⟨w_1,w_2,w_3⟩$ that satisfy the equation $\displaystyle ⟨1,1,1⟩×w=⟨−1,−1,2⟩.$

Answer: $\displaystyle w=⟨w_3−1,w_3+1,w_3⟩,$ where $\displaystyle w_3$ is any real number

52) Solve the equation $\displaystyle w×⟨1,0,−1⟩=⟨3,0,3⟩,$ where $\displaystyle w=⟨w_1,w_2,w_3⟩$ is a nonzero vector with a magnitude of $\displaystyle 3$.

53) [T] A mechanic uses a 12-in. wrench to turn a bolt. The wrench makes a $\displaystyle 30°$ angle with the horizontal. If the mechanic applies a vertical force of $\displaystyle 10$ lb on the wrench handle, what is the magnitude of the torque at point $\displaystyle P$ (see the following figure)? Express the answer in foot-pounds rounded to two decimal places.

$This figure is the image of an open-end wrench. The lower portion of the wrench is at point P. The wrench has a length of “12 I n.” The angle the wrench makes with a horizontal line from P is 30 degrees. At the top of the wrench is a downward vertical vector labeled “10 l b.”$

Answer: 8.66 ft-lb

54) [T] A boy applies the brakes on a bicycle by applying a downward force of 20 lb on the pedal when the 6-in. crank makes a $\displaystyle 40°$ angle with the horizontal (see the following figure). Find the torque at point $\displaystyle P$. Express your answer in foot-pounds rounded to two decimal places.

$This figure shows the pedals, cranks, and chain of a bicycle. The distance along the crank to the top pedal is 6 in. The angle of the crank is 40 degrees with the horizontal, measured toward the rear. The top pedal has a downward vector labeled “20 lb”.$

55) [T] Find the magnitude of the force that needs to be applied to the end of a 20-cm wrench located on the positive direction of the y-axis if the force is applied in the direction $\displaystyle ⟨0,1,−2⟩$ and it produces a $\displaystyle 100$ N·m torque to the bolt located at the origin.

Answer: 250 N

56) [T] What is the magnitude of the force required to be applied to the end of a 1-ft wrench at an angle of $\displaystyle 35°$ to produce a torque of $\displaystyle 20$ N·m?

57) [T] The force vector $\displaystyle F$ acting on a proton with an electric charge of $\displaystyle 1.6×10^{−19}C$ (in coulombs) moving in a magnetic field $\displaystyle B$ where the velocity vector $\displaystyle v$ is given by $\displaystyle F=1.6×10^{−19}(v×B)$ (here, $\displaystyle v$ is expressed in meters per second, $\displaystyle B$ is in tesla [T], and $\displaystyle F$ is in newtons [N]). Find the force that acts on a proton that moves in the xy-plane at velocity $\displaystyle v=10^5i+10^5j$ (in meters per second) in a magnetic field given by $\displaystyle B=0.3j$.

Answer: $\displaystyle F=4.8×10^{−15}kN$

58) [T] The force vector $\displaystyle F$ acting on a proton with an electric charge of $\displaystyle 1.6×10^{−19}C$ moving in a magnetic field $\displaystyle B$ where the velocity vector $\displaystyle v$ is given by $\displaystyle F=1.6×10^{−19}(v×B)$ (here, $\displaystyle v$ is expressed in meters per second, $\displaystyle B$ in $\displaystyle T$, and $\displaystyle F$ in $\displaystyle N$). If the magnitude of force $\displaystyle F$ acting on a proton is $\displaystyle 5.9×10^{−17} N$ and the proton is moving at the speed of 300 m/sec in magnetic field $\displaystyle B$ of magnitude 2.4 T, find the angle between velocity vector $\displaystyle v$ of the proton and magnetic field $\displaystyle B$. Express the answer in degrees rounded to the nearest integer.

60) [T] Consider $\displaystyle r(t)=⟨cost,sint,2t⟩$ the position vector of a particle at time $\displaystyle t∈[0,30]$, where the components of $\displaystyle r$ are expressed in centimeters and time in seconds. Let $\displaystyle \vec{OP}$ be the position vector of the particle after $\displaystyle 1$ sec.

a. Determine unit vector $\displaystyle B(t)$ (called the binormal unit vector) that has the direction of cross product vector $\displaystyle v(t)×a(t),$ where $\displaystyle v(t)$ and $\displaystyle a(t)$ are the instantaneous velocity vector and, respectively, the acceleration vector of the particle after $\displaystyle t$ seconds.

b. Use a CAS to visualize vectors $\displaystyle v(1), a(1),$ and $\displaystyle B(1)$ as vectors starting at point $\displaystyle P$ along with the path of the particle.

Answer

$\displaystyle a. B(t)=⟨\frac{2sint}{\sqrt{5}},−\frac{2cost}{\sqrt{5}},\frac{1}{\sqrt{5}}⟩;$

$This figure is the first octant of the 3-dimensional coordinate system. There is a curve sketched that is increasing. On the curve is a point labeled “P.” At P there is a tangent vector to the curve labeled “v(1).” Also from P there is a vector towards the inside of the curve labeled “a(1).” Finally, there is a vector from P labeled “B(1)” pointing towards the z-axis.$

61) A solar panel is mounted on the roof of a house. The panel may be regarded as positioned at the points of coordinates (in meters) $\displaystyle A(8,0,0), B(8,18,0), C(0,18,8),$ and $\displaystyle D(0,0,8)$ (see the following figure).

$This figure shows a rectangular set of solar panels on a roof. The corners are labeled “A, B, C, D.” Also there is a vector drawn from A to D. There is another vector along the bottom of the rectangle from A to B.$

a. Find vector $\displaystyle n=\vec{AB}×\vec{AD}$ perpendicular to the surface of the solar panels. Express the answer using standard unit vectors.

b. Assume unit vector $\displaystyle s=\frac{1}{\sqrt{3}}i+\frac{1}{\sqrt{3}}j+\frac{1}{\sqrt{3}}k$ points toward the Sun at a particular time of the day and the flow of solar energy is $\displaystyle F=900s$ (in watts per square meter [$\displaystyle W/m^2$]). Find the predicted amount of electrical power the panel can produce, which is given by the dot product of vectors $\displaystyle F$ and $\displaystyle n$ (expressed in watts).

c. Determine the angle of elevation of the Sun above the solar panel. Express the answer in degrees rounded to the nearest whole number. (Hint: The angle between vectors $\displaystyle n$ and $\displaystyle s$ and the angle of elevation are complementary.)

Exercise \(\PageIndex{1}\)

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Exercise \(\PageIndex{9}\)

Exercise \(\PageIndex{10}\)