1.E: Vectors in Euclidian Space (Exercises)

1.1: Introduction

A

1.1.1. Calculate the magnitudes of the following vectors:\(\text{(a) } \textbf{v} = (2,-1) \quad \text{(b) } \textbf{v} = (2,-1,0) \quad \text{(c) } \textbf{v} = (3,2,-2) \quad \text{(d) }\textbf{v} = (0,0,1) \quad \text{ (e) } \textbf{v} = (6,4,-4)\)

1.1.2. For the points \(P =(1,-1,1)\), \(Q=(2,-2,2)\), \(R=(2,0,1)\), \(S=(3,-1,2)\), does \(\overrightarrow{PQ} = \overrightarrow{RS}\)?

1.1.3 For the points \(P =(0,0,0)\), \(Q=(1,3,2)\), \(R=(1,0,1)\), \(S=(2,3,4)\), does \(\overrightarrow{PQ} = \overrightarrow{RS}\)?

B

1.1.4 Let \(\textbf{v} = (1,0,0)\) and \(\textbf{w} = (a,0,0)\) be vectors in \(\mathbb{R}^{3}\). Show that \(\norm{\textbf{w}} = |a| \,\norm{\textbf{v}}\).

1.1.5 Let \(\textbf{v} = (a,b,c)\) and \(\textbf{w} = (3a,3b,3c)\) be vectors in \(\mathbb{R}^{3}\). Show that \(\norm{\textbf{w}} = 3 \,\norm{\textbf{v}}\).

C

Figure 1.1.9

1.1.6 Though we will see a simple proof of Theorem 1.1 in the next section, it is possible to prove it using methods similar to those in the proof of Theorem 1.2. Prove the special case of Theorem 1.1 where the points \(P = (x_{1}, y_{1}, z_{1})\) and \(Q = x_{2}, y_{2}, z_{2})\) satisfy the following conditions:
\(x_{2} > x_{1} > 0\), \(y_{2} > y_{1} > 0\), and \(z_{2} > 1 > 0\).
\(\textit{Hint: Think of Case 4 in the proof of Theorem (1.2), and consider Figure 1.1.9.}\)

1.2: Vector Algebra

A

1.2.1 Let \(\textbf{v} = (-1,5,-2)\) and \(\textbf{w} = (3,1,1)\).

• Find \(\textbf{v} - \textbf{w}\).
• Find \(\textbf{v} + \textbf{w}\).
• Find \(\frac{\textbf{v}}{\norm{\textbf{v}}}\).
• Find \(\norm{\frac{1}{2}(\textbf{v} - \textbf{w})}\).
• Find \(\norm{\frac{1}{2}(\textbf{v} + \textbf{w})}\).
• Find \(-2\,\textbf{v} + 4\,\textbf{w}\).
• Find \(\textbf{v} - 2\,\textbf{w}\).
• Find the vector \(\textbf{u}\) such that \(\textbf{u} + \textbf{v} + \textbf{w} = \textbf{i}\).
• Find the vector \(\textbf{u}\) such that \(\textbf{u} + \textbf{v} + \textbf{w} = 2\,\textbf{j} + \textbf{k}\).
• Is there a scalar \(m\) such that \(m(\textbf{v} + 2\,\textbf{w}) = \textbf{k}\)? If so, find it.

1.2.2. For the vectors \(\textbf{v}\) and \(\textbf{w}\) from Exercise 1, is \(\norm{\textbf{v} - \textbf{w}} = \norm{\textbf{v}} - \norm{\textbf{w}}\)? If not, which quantity is larger?

1.2.3. For the vectors \(\textbf{v}\) and \(\textbf{w}\) from Exercise 1, is \(\norm{\textbf{v} + \textbf{w}} = \norm{\textbf{v}} + \norm{\textbf{w}}\)? If not, which quantity is larger?

B

1.2.4. Prove Theorem 1.5 (f) for \(\mathbb{R}^{3}\).

1.2.5. Prove Theorem 1.5 (g) for \(\mathbb{R}^{3}\).

C

1.2.6. We know that every vector in \(\mathbb{R}^{3}\) can be written as a scalar combination of the vectors \(\textbf{i}\), \(\textbf{j}\), and \(\textbf{k}\). Can every vector in \(\mathbb{R}^{3}\) be written as a scalar combination of just \(\textbf{i}\) and \(\textbf{j}\), i.e. for any vector \(\textbf{v}\) in \(\mathbb{R}^{3}\), are there scalars \(m\), \(n\) such that \(\textbf{v} = m\,\textbf{i} + n\,\textbf{j}\)? Justify your answer.

1.3: Dot Product

A

1.3.1. Let \(\textbf{v} = (5,1,-2)\) and \(\textbf{w} = (4,-4,3)\). Calculate \(\textbf{v} \cdot \textbf{w}\).

1.3.2. Let \(\textbf{v} = -3\,\textbf{i} - 2\,\textbf{j} - \textbf{k}\) and \(\textbf{w} = 6\,\textbf{i} + 4\,\textbf{j} + 2\,\textbf{k}\). Calculate \(\textbf{v} \cdot \textbf{w}\).

For Exercises 3-8, find the angle \(\theta\) between the vectors \(\textbf{v}\) and \(\textbf{w}\).

1.3.3. \(\textbf{v} = (5,1,-2)\), \(\textbf{w} = (4,-4,3)\)

1.3.4. \(\textbf{v} = (7,2,-10)\), \(\textbf{w} = (2,6,4)\)

1.3.5. \(\textbf{v} = (2,1,4)\), \(\textbf{w} = (1,-2,0)\)

1.3.6. \(\textbf{v} = (4,2,-1)\), \(\textbf{w} = (8,4,-2)\)

1.3.7. \(\textbf{v} = -\,\textbf{i} + 2\,\textbf{j} + \textbf{k}\), \(\textbf{w} = -3\,\textbf{i} + 6\,\textbf{j} + 3\,\textbf{k}\)

1.3.8. \(\textbf{v} = \textbf{i}\), \(\textbf{w} = 3\,\textbf{i} + 2\,\textbf{j} + 4\textbf{k}\)

1.3.9. Let \(\textbf{v} = (8,4,3)\) and \(\textbf{w} = (-2,1,4)\). Is \(\textbf{v} \perp \textbf{w}\)? Justify your answer.

1.3.10. Let \(\textbf{v} = (6,0,4)\) and \(\textbf{w} = (0,2,-1)\). Is \(\textbf{v} \perp \textbf{w}\)? Justify your answer.

1.3.11. For \(\textbf{v}\), \(\textbf{w}\) from Exercise 5, verify the Cauchy-Schwarz Inequality \(|\textbf{v} \cdot \textbf{w}| \le \norm{\textbf{v}}\,\norm{\textbf{w}}\).

1.3.12. For \(\textbf{v}\), \(\textbf{w}\) from Exercise 6, verify the Cauchy-Schwarz Inequality \(|\textbf{v} \cdot \textbf{w}| \le \norm{\textbf{v}}\,\norm{\textbf{w}}\).

1.3.13. For \(\textbf{v}\), \(\textbf{w}\) from Exercise 5, verify the Triangle Inequality \(\norm{\textbf{v} + \textbf{w}} \le \norm{\textbf{v}} + \norm{\textbf{w}}\).

1.3.14. For \(\textbf{v}\), \(\textbf{w}\) from Exercise 6, verify the Triangle Inequality \(\norm{\textbf{v} + \textbf{w}} \le \norm{\textbf{v}} + \norm{\textbf{w}}\).

B

Note: Consider only vectors in \(\mathbb{R}^{3}\) for Exercises 15-25.

1.3.15. Prove Theorem 1.9 (a).

1.3.16. Prove Theorem 1.9 (b).

1.3.17. Prove Theorem 1.9 (c).

1.3.18. Prove Theorem 1.9 (d).

1.3.19. Prove Theorem 1.9 (e).

1.3.20. Prove Theorem 1.10 (a).

1.3.21. Prove or give a counterexample: If \(\textbf{u} \cdot \textbf{v} = \textbf{u} \cdot \textbf{w}\), then \(\textbf{v} =\textbf{w}\).

C

1.3.22. Prove or give a counterexample: If \(\textbf{v} \cdot \textbf{w} = 0\) for all \(\textbf{v}\), then \(\textbf{w} =\textbf{0}\).

1.3.23. Prove or give a counterexample: If \(\textbf{u} \cdot \textbf{v} = \textbf{u} \cdot \textbf{w}\) for all \(\textbf{u}\), then \(\textbf{v} = \textbf{w}\).

1.3.24. Prove that \(|\norm{\textbf{v}} - \norm{\textbf{w}}| \le \norm{\textbf{v} - \textbf{w}}\) for all \(\textbf{v}, \textbf{w}\).

Figure 1.3.5

1.3.25. For nonzero vectors \(\textbf{v}\) and \(\textbf{w}\), the \(\textit{projection}\) of \(\textbf{v}\) onto \(\textbf{w}\) (sometimes written as \(proj_{\textbf{w}}\textbf{v}\)) is the vector \(\textbf{u}\) along the same line \(L\) as \(\textbf{w}\) whose terminal point is obtained by dropping a perpendicular line from the terminal point of \(\textbf{v}\) to \(L\) (see Figure 1.3.5). Show that

$$\norm{\textbf{u}} = \frac{|\textbf{v} \cdot \textbf{w}|}{\norm{\textbf{w}}}.$$

\(\textit{(Hint: Consider the angle between}\) \(\textbf{v}\) \(\textit{and}\) \(\textbf{w}\).\(\textit{)}\)

1.3.26. Let \(\alpha\), \(\beta\), and \(\gamma\) be the angles between a nonzero vector \(\textbf{v}\) in \(\mathbb{R}^{3}\) and the vectors \(\textbf{i}\), \(\textbf{j}\), and \(\textbf{k}\), respectively. Show that \(\cos^{2} \alpha + \cos^{2} \beta + \cos^{2} \gamma = 1\).

(Note: \(\alpha\), \(\beta\), \(\gamma\) are often called the \(\textit{direction angles}\) of \(\textbf{v}\), and \(\cos \alpha\), \(\cos \beta\), \(\cos \gamma\) are called the \(\textit{direction cosines}\).)

1.4: Cross Product

A

For Exercises 1-6, calculate \(\textbf{v} \times \textbf{w}\).
1.4.1. \(\textbf{v} = (5,1,-2)\), \(\textbf{w} = (4,-4,3)\)
1.4.2. \(\textbf{v} = (7,2,-10)\), \(\textbf{w} = (2,6,4)\)
1.4.3. \(\textbf{v} = (2,1,4)\), \(\textbf{w} = (1,-2,0)\)
1.4.4. \(\textbf{v} = (1,3,2)\), \(\textbf{w} = (7,2,-10)\)
1.4.5. \(\textbf{v} = -\,\textbf{i} + 2\,\textbf{j} + \textbf{k}\), \(\textbf{w} = -3\,\textbf{i} + 6\,\textbf{j} + 3\,\textbf{k}\)
1.4.6. \(\textbf{v} = \textbf{i}\), \(\textbf{w} = 3\,\textbf{i} + 2\,\textbf{j} + 4\textbf{k}\)

For Exercises 7-8, calculate the area of the triangle \(\triangle PQR\).
1.4.7. \(P = (5,1,-2)\), \(Q = (4,-4,3)\), \(R =(2,4,0)\)
1.4.8. \(P = (4,0,2)\), \(Q = (2,1,5)\), \(R =(-1,0,-1)\)

For Exercises 9-10, calculate the area of the parallelogram \(PQRS\).
1.4.9. \(P = (2,1,3)\), \(Q = (1,4,5)\), \(R = (2,5,3)\), \(S = (3,2,1)\)
1.4.10. \(P = (-2,-2)\), \(Q = (1,4)\), \(R = (6,6)\), \(S = (3,0)\)

For Exercises 11-12, find the volume of the parallelepiped with adjacent sides \(\textbf{u}, \textbf{v}, \textbf{w}\).
1.4.11. \(\textbf{u} = (1,1,3)\), \(\textbf{v} = (2,1,4)\), \(\textbf{w} = (5,1,-2)\)
1.4.12. \(\textbf{u} = (1,3,2)\), \(\textbf{v} = (7,2,-10)\), \(\textbf{w} = (1,0,1)\)

For Exercises 13-14, calculate \(\textbf{u} \cdot (\textbf{v} \times \textbf{w})\) and \(\textbf{u} \times (\textbf{v} \times \textbf{w})\).
1.4.13. \(\textbf{u} = (1,1,1)\), \(\textbf{v} = (3,0,2)\), \(\textbf{w} = (2,2,2)\)
1.4.14. \(\textbf{u} = (1,0,2)\), \(\textbf{v} = (-1,0,3)\), \(\textbf{w} = (2,0,-2)\)

1.4.15. Calculate \((\textbf{u} \times \textbf{v}) \cdot (\textbf{w} \times \textbf{z})\) for \(\textbf{u} = (1,1,1)\), \(\textbf{v} = (3,0,2)\), \(\textbf{w} = (2,2,2)\), \(\textbf{z} = (2,1,4)\).

B

1.4.16. If \(\textbf{v}\) and \(\textbf{w}\) are unit vectors in \(\mathbb{R}^{3}\), under what condition(s) would \(\textbf{v} \times \textbf{w}\) also be a unit vector in \(\mathbb{R}^{3}\;\)? Justify your answer.

1.4.17. Show that if \(\textbf{v} \times \textbf{w} = \textbf{0}\) for all \(\textbf{w}\) in \(\mathbb{R}^{3}\), then \(\textbf{v} = \textbf{0}\).

1.4.18. Prove Theorem 1.14(b).

1.4.19. Prove Theorem 1.14(c).

1.4.20. Prove Theorem 1.14(d).

1.4.21. Prove Theorem 1.14(e).

1.4.22. Prove Theorem 1.14(f).

1.4.23. Prove Theorem 1.16.

1.4.24. Prove Theorem 1.17. (Hint: Expand both sides of the equation.)

1.4.25. Prove the following for all vectors \(\textbf{v}, \,\textbf{w}\text{ in }\mathbb{R}^ 3\):

(a) \(\norm{\textbf{v} \times \textbf{w}}^{2} + | \textbf{v} \cdot \textbf{w}|^{2} = \norm{\textbf{v}}^2 \, \norm{\textbf{w}}^2 \)

(b) If \(\textbf{v} \cdot \textbf{w} = 0\) and \(\textbf{v} \times \textbf{w} = \textbf{0}\), then \(\textbf{v} = \textbf{0}\) or \(\textbf{w} = \textbf{0}\).

C

1.4.26. Prove that in Example 1.8 the formula for the area of the triangle \(\triangle PQR\) yields the same value no matter which two adjacent sides are chosen. To do this, show that \(\frac{1}{2}\,\norm{\textbf{u} \times (-\textbf{w})} = \frac{1}{2}\,\norm{\textbf{v} \times \textbf{w}}\), where \(\textbf{u} = PR\), \(-\textbf{w} = PQ\), and \(\textbf{v} = QR\), \(\textbf{w} = QP\) as before. Similarly, show that \(\frac{1}{2}\,\norm{(-\textbf{u}) \times (-\textbf{v})} = \frac{1}{2}\,\norm{\textbf{v} \times \textbf{w}}\), where \(-\textbf{u} = RP\) and \(-\textbf{v} = RQ\).

1.4.27. Consider the vector equation \(\textbf{a} \times \textbf{x} = \textbf{b}\) in \(\mathbb{R}^{3}\), where \(\textbf{a} \ne \textbf{0}\). Show that:

• \(\textbf{a} \cdot \textbf{b} = 0\)
• \(\textbf{x} = \dfrac{\textbf{b} \times \textbf{a}}{\norm{\textbf{a}}^{2}} + k \textbf{a}\) is a solution to the equation, for any scalar \(k\).

1.4.28. Prove the \(\textit{Jacobi identity}\):
$$\textbf{u} \times (\textbf{v} \times \textbf{w}) + \textbf{v} \times (\textbf{w} \times \textbf{u}) + \textbf{w} \times (\textbf{u} \times \textbf{v}) = \textbf{0}$$

1.4.29. Show that \(\textbf{u}, \textbf{v}, \textbf{w}\) lie in the same plane in \(\mathbb{R}^{3}\) if and only if \(\textbf{u} \cdot (\textbf{v} \times \textbf{w}) = 0\).

1.4.30. For all vectors \(\textbf{u}, \textbf{v}, \textbf{w}, \textbf{z}\) in \(\mathbb{R}^{3}\), show that
\[(\textbf{u} \times \textbf{v}) \times (\textbf{w} \times \textbf{z}) =
(\textbf{z} \cdot (\textbf{u} \times \textbf{v}))\textbf{w} -
(\textbf{w} \cdot (\textbf{u} \times \textbf{v}))\textbf{z}\]

and that

\[(\textbf{u} \times \textbf{v}) \times (\textbf{w} \times \textbf{z}) =
(\textbf{u} \cdot (\textbf{w} \times \textbf{z}))\textbf{v} -
(\textbf{v} \cdot (\textbf{w} \times \textbf{z}))\textbf{u}\]

Why do both equations make sense geometrically?

1.5: Lines and Planes

A

For Exercises 1-4, write the line \(L\) through the point \(P\) and parallel to the vector \(\mathbf{v}\) in the following forms: (a) vector, (b) parametric, and (c) symmetric.

1.5.1. \(P = (2,3,−2), \, \mathbf{v} = (5,4,−3)\)

1.5.2. \(P = (3,−1,2), \, \mathbf{v} = (2,8,1)\)

1.5.3. \(P= (2,1,3), \, \mathbf{v} = (1,0,1)\)

1.5.4. \(P = (0,0,0), \, \mathbf{v} = (7,2,−10)\)

For Exercises 5-6, write the line \(L\) through the points \(P_1 \text{ and }P_2\) in parametric form.

1.5.5. \(P_1 = (1,−2,−3),\, P_2 = (3,5,5)\)

1.5.6. \(P_1 = (4,1,5),\, P_2 = (−2,1,3)\)

For Exercises 7-8, find the distance \(d\) from the point \(P\) to the line \(L\).

1.5.7. \(P = (1,−1,−1),\, L : x = −2−2t,\, y = 4t,\, z = 7+ t\)

1.5.8. \(P = (0,0,0),\, L : x = 3+2t,\, y = 4+3t,\, z = 5+4t\)

For Exercises 9-10, find the point of intersection (if any) of the given lines.

1.5.9. \(x = 7+3s,\, y = −4−3s,\, z = −7−5s \text{ and }x = 1+6t,\, y = 2+ t,\, z = 3−2t\)

1.5.10. \(\dfrac{x−6}{ 4} = y+3 = z \text{ and }\dfrac{x−11}{ 3} = \dfrac{y−14}{ −6} = \dfrac{z +9}{2}\)

For Exercises 11-12, write the normal form of the plane \(P\) containing the point \(Q\) and perpendicular to the vector \(\textbf{n}\).

1.5.11. \(Q = (5,1,−2),\, \textbf{n} = (4,−4,3)\)

1.5.12. \(Q = (6,−2,0),\, \textbf{n} = (2,6,4)\)

For Exercises 13-14, write the normal form of the plane containing the given points.

1.5.13. \((1,0,3),\, (1,2,−1),\, (6,1,6)\)

1.5.14. \((−3,1,−3),\, (4,−4,3),\, (0,0,1)\)

1.5.15. Write the normal form of the plane containing the lines from Exercise 9.

1.5.16. Write the normal form of the plane containing the lines from Exercise 10.

For Exercises 17-18, find the distance \(D\) from the point \(Q\) to the plane \(P\).

1.5.17. \(Q = (4,1,2),\, P : 3x− y−5z +8 = 0 \)

1.5.18. \(Q = (0,2,0),\, P : −5x+2y−7z +1 = 0\)

For Exercises 19-20, find the line of intersection (if any) of the given planes.

1.5.19. \(x+3y+2z −6 = 0,\, 2x− y+ z +2 = 0 \)

1.5.20. \(3x+ y−5z = 0,\, x+2y+ z +4 = 0\)

B

1.5.21. Find the point(s) of intersection (if any) of the line \(\dfrac{x−6}{ 4} = y + 3 = z\) with the plane \(x+3y+2z −6 = 0\). (Hint: Put the equations of the line into the equation of the plane.)

1.6: Surfaces

A

For Exercises 1-4, determine if the given equation describes a sphere. If so, find its radius and center.

1.6.1. \(x^ 2 + y^ 2 + z^ 2 −4x−6y−10z +37 = 0 \)

1.6.2. \(x^ 2 + y^ 2 + z^ 2 +2x−2y−8z +19 = 0\)

1.6.3. \(2x^ 2 +2y^ 2 +2z^ 2 +4x+4y+4z −44 = 0 \)

1.6.4. \(x^ 2 + y^ 2 − z^ 2 +12x+2y−4z +32 = 0\)

1.6.5. Find the point(s) of intersection of the sphere \((x − 3)^2 + (y + 1)^2 + (z − 3)^2 = 9\) and the line \(x = −1+2t,\, y = −2−3t,\, z = 3+ t\).

B

1.6.6. Find the intersection of the spheres \(x^ 2 + y^ 2 + z^ 2 = 9 \text{ and }(x−4)^2 +(y+2)^2 +(z −4)^2 = 9\).

1.6.7. Find the intersection of the sphere \(x^ 2 + y^ 2 + z^ 2 = 9 \text{ and the cylinder }x^ 2 + y^ 2 = 4\).

1.6.8. Find the trace of the hyperboloid of one sheet \(\dfrac{x^ 2}{ a^ 2} + \dfrac{y^ 2}{ b^ 2} − \dfrac{z^ 2}{ c^ 2} = 1\) in the plane \(x = a\), and the trace in the plane \(y = b\).

1.6.9. Find the trace of the hyperbolic paraboloid \(\frac{x^ 2}{ a^ 2} − \frac{y^ 2}{ b^ 2} = \frac{z}{ c}\) in the \(x y\)-plane.

C

1.6.10. It can be shown that any four noncoplanar points (i.e. points that do not lie in the same plane) determine a sphere. Find the equation of the sphere that passes through the points \((0,0,0),\, (0,0,2),\, (1,−4,3) \text{ and }(0,−1,3)\). (Hint: Equation (1.31))

1.6.11. Show that the hyperboloid of one sheet is a doubly ruled surface, i.e. each point on the surface is on two lines lying entirely on the surface. (Hint: Write equation (1.35) as \(\frac{x^ 2}{ a^ 2} − \frac{z^ 2}{ c^ 2} = 1− \frac{y^ 2}{ b^ 2}\) , factor each side. Recall that two planes intersect in a line.)

1.6.12. Show that the hyperbolic paraboloid is a doubly ruled surface. (Hint: Exercise 11)

Let \(S\) be the sphere with radius 1 centered at \((0,0,1)\), and let \(S^∗\) be \(S\) without the “north pole” point \((0,0,2)\). Let \((a,b, c)\) be an arbitrary point on \(S^∗\) . Then the line passing through \((0,0,2) \text{ and }(a,b, c)\) intersects the \(x y\)-plane at some point \((x, y,0)\), as in Figure 1.6.10. Find this point \((x, y,0)\) in terms of \(a, b \text{ and }c\).

Figure 1.6.10

(Note: Every point in the \(x y\)-plane can be matched with a point on \(S^ ∗\) , and vice versa, in this manner. This method is called stereographic projection, which essentially identifies all of \(\mathbb{R}^ 2\) with a “punctured” sphere.)

1.7: Curvilinear Coordinates

A

For Exercises 1-4, find the (a) cylindrical and (b) spherical coordinates of the point whose Cartesian coordinates are given.

1.7.1. \((2,2 \sqrt{ 3},−1)\)

1.7.2. \((−5,5,6)\)

1.7.3. \(( \sqrt{ 21},−\sqrt{ 7},0)\)

1.7.4. \((0,\sqrt{ 2},2)\)

For Exercises 5-7, write the given equation in (a) cylindrical and (b) spherical coordinates.

1.7.5. \(x^ 2 + y^ 2 + z^ 2 = 25\)

1.7.6. \(x^ 2 + y^ 2 = 2y\)

1.7.7. \(x^ 2 + y^ 2 +9z^ 2 = 36\)

B

1.7.8. Describe the intersection of the surfaces whose equations in spherical coordinates are \(θ = \frac{π}{ 2} \text{ and }φ = \frac{π}{ 4} \).

1.7.9. Show that for \(a \neq 0\), the equation \(ρ = 2a\sin φ \cos θ\) in spherical coordinates describes a sphere centered at \((a,0,0) \text{ with radius }|a|\).

C

1.7.10. Let \(P = (a,θ,φ)\) be a point in spherical coordinates, with \(a > 0 \text{ and }0 < φ < π\). Then \(P\) lies on the sphere \(ρ = a\). Since \(0 < φ < π\), the line segment from the origin to \(P\) can be extended to intersect the cylinder given by \(r = a\) (in cylindrical coordinates). Find the cylindrical coordinates of that point of intersection.

1.7.11. Let \(P_1 \text{ and }P_2\) be points whose spherical coordinates are \((ρ_1 ,θ_1 ,φ_1) \text{ and }(ρ_2 ,θ_2 ,φ_2)\), respectively. Let \(\mathbf{v}_1\) be the vector from the origin to \(P_1\) , and let \(\mathbf{v}_2\) be the vector from the origin to \(P_2\) . For the angle \(γ\) between \(\mathbf{v}_1 \text{ and }\mathbf{v}_2\) , show that

\[\cos γ = \cos φ_1 \cos φ_2 +\sin φ_1 \sin φ_2 \cos (θ_2 −θ_1 ).\]

This formula is used in electrodynamics to prove the addition theorem for spherical harmonics, which provides a general expression for the electrostatic potential at a point due to a unit charge. See pp. 100-102 in JACKSON.

1.7.12. Show that the distance d between the points \(P_1 \text{ and }P_2\) with cylindrical coordinates \((r_1 ,θ_1 , z_1) \text{ and }(r_2 ,θ_2 , z_2)\), respectively, is

\[d = \sqrt{ r_1^2 + r_2^2 −2r_1 r_2 \cos (θ_2 −θ_1 )+(z_2 − z_1)^ 2} .\]

1.7.13. Show that the distance \(d\) between the points \(P_1 \text{ and }P_2\) with spherical coordinates \((ρ_1 ,θ_1 ,φ_1) \text{ and }(ρ_2 ,θ_2 ,φ_2)\), respectively, is

\[d = \sqrt{ ρ_1^2 +ρ_2^2 −2ρ_1 ρ_2[\sin φ_1 \sin φ_2 \cos (θ_2 −θ_1 )+\cos φ_1 \cos φ_2]}.\]

1.8: Vector-Valued Functions

A

For Exercises 1-4, calculate \(f ′ (t)\) and find the tangent line at \(f(0)\).

1.8.1. \(\textbf{f}(t) = (t+1,t^ 2 +1,t^ 3 +1)\)

1.8.2. \(\textbf{f}(t) = (e^ t +1, e^ {2t} +1, e^ {t^ 2} +1)\)

1.8.3. \(\textbf{f}(t) = (\cos 2t,\sin 2t,t)\)

1.8.4. \(\textbf{f}(t) = (\sin 2t,2\sin^2 t,2\cos t)\)

For Exercises 5-6, find the velocity \(\textbf{v}(t) \text{ and acceleration }\textbf{a}(t) \text{ of an object with the given position vector }\textbf{r}(t)\).

1.8.5. \(\textbf{r}(t) = (t,t−\sin t,1−\cos t)\)

1.8.6. \(\textbf{r}(t) = (3\cos t,2\sin t,1)\)

B

1.8.7. Let \(f(t) = \left ( \frac{\cos t}{ \sqrt{ 1+ a^ 2 t^ 2}} , \frac{\sin t}{\sqrt{ 1+ a^ 2 t^ 2}} , \frac{−at}{ \sqrt{ 1+ a^ 2 t^ 2}} \right ) , \text{ with }a \neq 0\).

(a) Show that \(\norm{\textbf{f}(t)} = 1 \text{ for all }t\)

(b) Show directly that \(\textbf{f} ′ (t)·\textbf{f}(t) = 0 \text{ for all } t\)

1.8.8. If \(\textbf{f} ′ (t) = 0 \text{ for all }t \text{ in some interval }(a,b),\text{ show that }\textbf{f}(t) \text{ is a constant vector in }(a,b).\)

1.8.9. For a constant vector \(\textbf{c} \neq 0,\) the function \(\textbf{f}(t) = t\textbf{c}\) represents a line parallel to \(\textbf{c}\).

(a) What kind of curve does \(\textbf{g}(t) = t^ 3\textbf{c}\) represent? Explain.

(b) What kind of curve does \(\textbf{h}(t) = e^ t\textbf{c}\) represent? Explain.

(c) Compare \(\textbf{f} ′ (0) \text{ and }\textbf{g} ′ (0)\). Given your answer to part (a), how do you explain the difference in the two derivatives?

1.8.10. Show that \(\frac{d}{ dt} \left ( \textbf{f} × \frac{df}{ dt} \right ) = \textbf{f} × \frac{d^ 2 f}{ dt^2}\) .

1.8.11. Let a particle of (constant) mass m have position vector \(\textbf{r}(t)\), velocity \(\textbf{v}(t)\), acceleration \(\textbf{a}(t)\) and momentum \(\textbf{p}(t)\) at time \(t\). The angular momentum \(\textbf{L}(t)\) of the particle with respect to the origin at time \(t\) is defined as \(\textbf{L}(t) = \textbf{r}(t)× \textbf{p}(t)\). If \(\textbf{F}(t)\) is the force acting on the particle at time \(t\), then define the torque \(\textbf{N}(t)\) acting on the particle with respect to the origin as \(\textbf{N}(t) = \textbf{r}(t)× \textbf{F}(t)\). Show that \(\textbf{L} ′ (t) = \textbf{N}(t)\).

1.8.12. Show that \(\frac{d}{ dt} (\textbf{f}· (\textbf{g} × \textbf{h})) = \frac{d\textbf{f}}{ dt} · (\textbf{g} × \textbf{h}) + \textbf{f}· \left ( \frac{d\textbf{g}}{ dt} × \textbf{h}\right ) + \textbf{f}· \left ( \textbf{g} × \frac{d\textbf{h}}{ dt} \right ) \) .

1.8.13. The Mean Value Theorem does not hold for vector-valued functions: Show that for \(\textbf{f}(t) = (\cos t,\sin t,t)\), there is no \(t\) in the interval \((0,2π)\) such that

\[\textbf{f} ′ (t) = \frac{\textbf{f}(2π)−\textbf{f}(0)}{ 2π−0} .\]

C

1.8.14. The Bézier curve \(\textbf{b}_0^3 (t)\) for four noncollinear points \(\textbf{b}_0 ,\, \textbf{b}_1 ,\, \textbf{b}_2 ,\, \textbf{b}_3\) in \(\mathbb{R}^ 3\) is defined by the following algorithm (going from the left column to the right):

\[\begin{split}&\textbf{b}_0^1 (t) = (1− t)\textbf{b}_0 + t\textbf{b}_1 \\ &\textbf{b}_1^1 (t) = (1− t)\textbf{b}_1 + t\textbf{b}_2 \\ &\textbf{b}_2^1 (t) = (1− t)\textbf{b}_2 + t\textbf{b}_3 \end{split} \quad \begin{split} &\textbf{b}_0^2 (t) = (1− t)\textbf{b}_0^1 (t)+ t\textbf{b}_1^1 (t) \\ &\textbf{b}_1^2 (t) = (1− t)\textbf{b}_1^1 (t)+ t\textbf{b}_2^1 (t) \\ & \\ \end{split} \quad \begin{split} &\textbf{b}_0^3 (t) = (1− t)\textbf{b}_0^2 (t)+ t\textbf{b}_1^2 (t) \\ & \\ & \\ \end{split}\]

1.8.15. Let \(\textbf{r}(t)\) be the position vector for a particle moving in \(\mathbb{R}^ 3\) . Show that

\[\frac{d}{ dt} (\textbf{r}× (\textbf{v} × \textbf{r})) = \norm{\textbf{r}}^2 \textbf{a}+(\textbf{r} · \textbf{v})\textbf{v}−(\norm{\textbf{v}}^2 +\textbf{r} · \textbf{a})\textbf{r}.\]

1.8.16. Let \(\textbf{r}(t)\) be the position vector in \(\mathbb{R}^ 3\) for a particle that moves with constant speed \(c > 0\) in a circle of radius \(a > 0\) in the \(x y\)-plane. Show that \(\textbf{a}(t)\) points in the opposite direction as \(\textbf{r}(t) \text{ for all }t\). (Hint: Use Example 1.37 to show that \(\textbf{r}(t) ⊥ \textbf{v}(t) \text{ and }\textbf{a}(t) ⊥ \textbf{v}(t)\), and hence \(\textbf{a}(t) ∥ \textbf{r}(t).\))

1.8.17. Prove Theorem 1.20(g).

1.9: Arc Length

A

For Exercises 1-3, calculate the arc length of \(\textbf{f}(t)\) over the given interval.

1.9.1. \(\textbf{f}(t) = (3\cos 2t,3\sin 2t,3t) \text{ on }[0,π/2]\)

1.9.2. \(\textbf{f}(t) = ((t^ 2 +1)\cos t,(t^ 2 +1)\sin t,2 \sqrt{ 2t}) \text{ on }[0,1]\)

1.9.3. \(\textbf{f}(t) = (2\cos 3t,2\sin 3t,2t^ {3/2})\text{ on }[0,1]\)

1.9.4. Parametrize the curve from Exercise 1 by arc length.

1.9.5. Parametrize the curve from Exercise 3 by arc length.

B

1.9.6. Let \(\textbf{f}(t) \text{ be a differentiable curve such that }\textbf{f}(t) \neq 0\) for all \(t\). Show that

\[ \frac{d}{ dt} \left ( \frac{\textbf{f}(t)}{ \norm{\textbf{f}(t)}} \right ) = \frac{\textbf{f}t) × (\textbf{f} ′ (t) × \textbf{f}(t))}{ \norm{\textbf{f}(t)}^ 3}. \]

Exercises 7-9 develop the moving frame field \(\textbf{T}, \textbf{N}, \textbf{B}\) at a point on a curve.

1.9.7. Let \(\textbf{f}(t)\) be a smooth curve such that \(\textbf{f} ′ (t) \neq 0\) for all \(t\). Then we can define the unit tangent vector \(\textbf{T}\) by

\[\textbf{T}(t) = \frac{\textbf{f} ′ (t)}{ \norm{\textbf{f} ′ (t)}} .\]

Show that

\[\textbf{T} ′ (t) = \frac{\textbf{f} ′ (t) × (\textbf{f} ′′(t) × \textbf{f} ′ (t))}{ \norm{\textbf{f} ′ (t)}^ 3} .\]

1.9.8. Continuing Exercise 7, assume that \(\textbf{f} ′ (t) \text{ and }\textbf{f} ′′(t)\) are not parallel. Then \(\textbf{T} ′ (t) \neq 0\) so we can define the unit principal normal vector \(\textbf{N}\) by

\[\textbf{N}(t) =\frac{ \textbf{T} ′ (t)}{ \norm{\textbf{T}′ (t)}} .\]

Show that

\[\textbf{N}(t) = \frac{\textbf{f} ′ (t) × (\textbf{f} ′′(t) × \textbf{f} ′ (t))}{ \norm{\textbf{f} ′ (t)}\, \norm{\textbf{f} ′′(t) × \textbf{f} ′ (t)}} .\]

1.9.9. Continuing Exercise 8, the unit binormal vector \(B\) is defined by

\[\textbf{B}(t) = \textbf{T}(t) × \textbf{N}(t).\]

Show that

\[\textbf{B}(t) = \frac{\textbf{f} ′ (t) × \textbf{f} ′′(t)}{ \norm{\textbf{f} ′ (t) × \textbf{f} ′′(t)}} .\]

Note: The vectors \(\textbf{T}(t), \,\textbf{N}(t) \text{ and }\textbf{B}(t)\) form a right-handed system of mutually perpendicular unit vectors (called orthonormal vectors) at each point on the curve \(\textbf{f}(t)\).

1.9.10. Continuing Exercise 9, the curvature \(κ\) is defined by

\[κ(t) = \frac{\norm{\textbf{T} ′ (t)}}{ \norm{\textbf{f} ′ (t)}} = \frac{\norm{\textbf{f} ′ (t) × (\textbf{f} ′′(t) × \textbf{f} ′ (t))}}{ \norm{\textbf{f} ′ (t)}^4} .\]

Show that

\[κ(t) = \frac{\textbf{f} ′ (t) × \textbf{f} ′′(t)}{ \norm{\textbf{f} ′ (t)}^3} \text{ and that }\textbf{T} ′ (t) = \norm{\textbf{f} ′ (t)}\,κ(t)\textbf{N}(t).\]

Note: \(κ(t)\) gives a sense of how “curved” the curve \(\textbf{f}(t)\) is at each point.

1.9.11. Find \(\textbf{T}, \,\textbf{N},\, \textbf{B}\) and \(κ\) at each point of the helix \(\textbf{f}(t) = (\cos t,\sin t,t).\)

1.9.12. Show that the arc length \(L\) of a curve whose spherical coordinates are \(ρ = ρ(t),\, θ = θ(t) \text{ and }φ = φ(t)\) for \(t\) in an interval \([a,b]\) is

\[L=\int_a^b \sqrt{ρ ′ (t)^ 2 +(ρ(t)^ 2 \sin^2φ(t))θ ′ (t)^ 2 +ρ(t)^ 2 φ′ (t)^ 2}\, dt\]

Contributors

• Michael Corral (Schoolcraft College). The content of this page is distributed under the terms of the GNU Free Documentation License, Version 1.2.