Chapter 11

Find the equation of the plane through \((2,5,1)\) that is parallel to the plane \(x-y+2 z=4\).

Find the sum \(\mathbf{u}+\mathbf{v}\), the difference \(\mathbf{u}-\mathbf{v}\), and the magnitudes \(\|\mathbf{u}\|\) and \(\|\mathbf{v}\|\) $$ \mathbf{u}=\langle 1,0,1\rangle, \mathbf{v}=\langle-5,0,0\rangle $$

Complete the squares to find the center and radius of the sphere whose equation is given (see Example 2). \(4 x^{2}+4 y^{2}+4 z^{2}-4 x+8 y+16 z-13=0\)

Name and sketch the graph of each of the following equations in three-space. $$ 9 x^{2}+25 y^{2}+9 z^{2}=225 $$

sketch the curve in the xy-plane. Then, for the given point, find the curvature and the radius of curvature. Finally, $$ y=x(x-4)^{2},(4,0) $$

\text { If } \mathbf{r}(t)=\sin 3 t \mathbf{i}-\cos 3 t \mathbf{j}, \text { find } D_{t}\left[\mathbf{r}(t) \cdot \mathbf{r}^{\prime}(t)\right]

Find the symmetric equations of the line through \((2,-4,5)\) that is parallel to the plane \(3 x+y-2 z=5\) and perpendicular to the line $$ \frac{x+8}{2}=\frac{y-5}{3}=\frac{z-1}{-1} $$

Find the equation of the plane through \((0,0,2)\) that is parallel to the plane \(x+y+z=1\).

Find all vectors perpendicular to both \(\langle 1,-2,-3\rangle\) and \(\langle-3,2,0\rangle\).

Find the sum \(\mathbf{u}+\mathbf{v}\), the difference \(\mathbf{u}-\mathbf{v}\), and the magnitudes \(\|\mathbf{u}\|\) and \(\|\mathbf{v}\|\) $$ \mathbf{u}=\langle 0.3,0.3,0.5\rangle, \mathbf{v}=\langle 2.2,1.3,-0.9\rangle $$

Complete the squares to find the center and radius of the sphere whose equation is given (see Example 2). \(x^{2}+y^{2}+z^{2}+8 x-4 y-22 z+77=0\)

Name and sketch the graph of each of the following equations in three-space. $$ 5 x+8 y-2 z=10 $$

Make the required change in the given equation. \(x^{2}+y^{2}=9\) to cylindrical coordinates

sketch the curve in the xy-plane. Then, for the given point, find the curvature and the radius of curvature. Finally, $$ y=\sin x,\left(\frac{\pi}{4}, \frac{\sqrt{2}}{2}\right) $$

\begin{array}{l} \text { 17. If } \mathbf{r}(t)=\sqrt{t-1} \mathbf{i}+\ln \left(2 t^{2}\right) \mathbf{j} \text { and } h(t)=e^{-3 t}, \text { find }\\\ D_{t}[h(t) \mathbf{r}(t)] . \end{array}

17\. Find the equation of the plane that contains the parallel lines $$ \left\\{\begin{array} { l } { x = - 2 + 2 t } \\ { y = 1 + 4 t \quad \text { and } } \\ { z = 2 - t } \end{array} \left\\{\begin{array}{l} x=2-2 t \\ y=3-4 t \\ z=1+t \end{array}\right.\right. $$

Find the equation of the plane through \((-1,-2,3)\) and perpendicular to both the planes \(x-3 y+2 z=7\) and \(2 x-2 y-z=-3\).

Find the angle \(A B C\) if the points are \(A(1,2,3)\), \(B(-4,5,6)\), and \(C(1,0,1)\)

Name and sketch the graph of each of the following equations in three-space. $$ y=\cos x $$

Make the required change in the given equation. \(x^{2}-y^{2}=25\) to cylindrical coordinates

sketch the curve in the xy-plane. Then, for the given point, find the curvature and the radius of curvature. Finally, $$ y^{2}=x-1,(1,0) $$

If \(\mathbf{r}(t)=\sin 2 t \mathbf{i}+\cosh t \mathbf{j}\) and \(h(t)=\ln (3 t-2)\), find \(D_{t}[h(t) \mathbf{r}(t)] .\)

Show that the lines $$ \frac{x-1}{-4}=\frac{y-2}{3}=\frac{z-4}{-2} $$ and $$ \frac{x-2}{-1}=\frac{y-1}{1}=\frac{z+2}{6} $$ intersect, and find the equation of the plane that they determine.

Find the equation of the plane through \((2,-1,4)\) that is perpendicular to both the planes \(x+y+z=2\) and \(x-y-z=4\).

Show that the triangle \(A B C\) is a right triangle if the vertices are \(A(6,3,3), B(3,1,-1)\), and \(C(-1,10,-2.5) .\) Hint: Check the angle at \(B\).

Mark pushes on a post in the direction \(\mathrm{S} 30^{\circ} \mathrm{E}\left(30^{\circ}\right.\) east of south) with a force of 60 pounds. Dan pushes on the same post in the direction \(\mathrm{S} 60^{\circ} \mathrm{W}\) with a force of 80 pounds. What are the magnitude and direction of the resultant force?

Name and sketch the graph of each of the following equations in three-space. $$ z=\sqrt{16-x^{2}-y^{2}} $$

Make the required change in the given equation. \(x^{2}+y^{2}+4 z^{2}=10\) to cylindrical coordinates

In Problems 19-30, find the velocity \(\mathbf{v}\), acceleration \(\mathbf{a}\), and speed \(s\) at the indicated time \(t=t_{1}\). \mathbf{r}(t)=4 t \mathbf{i}+5\left(t^{2}-1\right) \mathbf{j}+2 t \mathbf{k} ; t_{1}=1

sketch the curve in the xy-plane. Then, for the given point, find the curvature and the radius of curvature. Finally, $$ y^{2}-4 x^{2}=20,(2,6) $$

Find the equation of the plane containing the line \(x=1+2 t, y=-1+3 t, z=4+t\) and the point \((1,-1,5) .\)

Find the equation of the plane through \((2,-3,2)\) and parallel to the plane of the vectors \(4 \mathbf{i}+3 \mathbf{j}-\mathbf{k}\) and \(2 \mathbf{i}-5 \mathbf{j}+6 \mathbf{k}\).

For what numbers \(c\) are \(\langle c, 6\rangle\) and \(\langle c,-4\rangle\) orthogonal?

A 300 -newton weight rests on a smooth (friction negligi- ble) inclined plane that makes an angle of \(30^{\circ}\) with the horizon- tal. What force parallel to the plane will just keep the weight from sliding down the plane? Hint: Consider the downward force of 300 newtons to be the sum of two forces, one parallel to the plane and one perpendicular to it.

Name and sketch the graph of each of the following equations in three-space. $$ z=\sqrt{x^{2}+y^{2}+1} $$

Make the required change in the given equation. \(x^{2}+y^{2}+4 z^{2}=10\) to spherical coordinates

Find the velocity \(\mathbf{v}\), acceleration \(\mathbf{a}\), and speed \(s\) at the indicated time \(t=t_{1}\). \mathbf{r}(t)=t \mathbf{i}+(t-1)^{2} \mathbf{j}+(t-3)^{3} \mathbf{k} ; t_{1}=0

Find the equation of the plane containing the line \(x=3 t, y=1+t, z=2 t\) and parallel to the intersection of the planes \(2 x-y+z=0\) and \(y+z+1=0\)

Find the equation of the plane through the origin that is perpendicular to the \(x y\) -plane and the plane \(3 x-2 y+z=4\).

For what numbers \(c\) are \(2 c \mathbf{i}-8 \mathbf{j}\) and \(3 \mathbf{i}+c \mathbf{j}\) orthogonal?

An object weighing \(258.5\) pounds is held in equilibrium by two ropes that make angles of \(27.34^{\circ}\) and \(39.22^{\circ}\), respectively, with the vertical. Find the magnitude of the force exerted on the object by each rope.

The graph of an equation in \(x, y\), and \(z\) is symmetric with respect to the \(x y\) -plane if replacing \(z\) by \(-z\) results in an equivalent equation. What condition leads to a graph that is symmetric with respect to each of the following? (a) \(y z\) -plane (b) \(z\) -axis (c) origin

Make the required change in the given equation. \(2 x^{2}+2 y^{2}-4 z^{2}=0\) to spherical coordinates

Find the velocity \(\mathbf{v}\), acceleration \(\mathbf{a}\), and speed \(s\) at the indicated time \(t=t_{1}\). $$ \mathbf{r}(t)=(1 / t) \mathbf{i}+\left(t^{2}-1\right)^{-1} \mathbf{j}+t^{5} \mathbf{k} ; t_{1}=2 $$

sketch the curve in the xy-plane. Then, for the given point, find the curvature and the radius of curvature. Finally, $$ y=\cos 2 x,\left(\frac{1}{6} \pi, \frac{1}{2}\right) $$

Find the equation of the plane through \((6,2,-1)\) and perpendicular to the line of intersection of the planes \(4 x-3 y+2 z+5=0\) and \(3 x+2 y-z+11=0\).

For what numbers \(c\) and \(d\) are \(\mathbf{u}=c \mathbf{i}+\mathbf{j}+\mathbf{k}\) and \(\mathbf{v}=2 \mathbf{j}+d \mathbf{k}\) orthogonal?

A wind with velocity 45 miles per hour is blowing in the direction N \(20^{\circ} \mathrm{W}\). An airplane that flies at 425 miles per hour in still air is supposed to fly straight north. How should the airplane be headed and how fast will it then be flying with respect to the ground?

What condition leads to a graph that is symmetric with respect to the following? (a) \(x z\) -plane (b) \(y\) -axis (c) \(x\) -axis

Find the velocity \(\mathbf{v}\), acceleration \(\mathbf{a}\), and speed \(s\) at the indicated time \(t=t_{1}\). $$ \mathbf{r}(t)=t^{6} \mathbf{i}+\left(6 t^{2}-5\right)^{6} \mathbf{j}+t \mathbf{k} ; t_{1}=1 $$