Chapter 11

In Problems 13-16, complete the squares to find the center and \(\mathrm{ra}\) dius of the sphere whose equation is given (see Example 2). $$ 4 x^{2}+4 y^{2}+4 z^{2}-4 x+8 y+16 z-13=0 $$

Find the parametric equations of the line through \((5,-3,4)\) that intersects the \(z\)-axis at a right angle.

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

For the three-dimensional vectors \(\mathbf{u}\) and \(\mathbf{v}\) in Problems 13-16, find the sum \(\mathbf{u}+\mathbf{v}\), the difference \(\mathbf{u}-\mathbf{v}\), and the magnitudes \(\|\mathbf{u}\|\) and \(\|\mathbf{v}\|\). $$ \mathbf{u}=\langle 0.3,0.3,0.5\rangle, \mathbf{v}=\langle 2.2,1.3,-0.9\rangle $$

In Problems 7-16, sketch the graph of the given cylindrical or spherical equation. \(r^{2} \cos ^{2} \theta+z^{2}=4\)

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

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

In Problems 13-16, complete the squares to find the center and \(\mathrm{ra}\) dius of the sphere whose equation is given (see Example 2). $$ x^{2}+y^{2}+z^{2}+8 x-4 y-22 z+77=0 $$

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

In Problems 17-30, 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, \(y=\sin x,\left(\frac{\pi}{4}, \frac{\sqrt{2}}{2}\right)\)

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

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. $$ 5 x+8 y-2 z=10 $$

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

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

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?

In Problems 17-30, 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 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 \(B\).

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

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

A 300 -newton weight rests on a smooth (friction negligible) inclined plane that makes an angle of \(30^{\circ}\) with the horizontal. 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.

In Problems 17-30, make the required change in the given equation. \(x^{2}+y^{2}+4 z^{2}=10\) to cylindrical coordinates

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

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

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

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

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.

In Problems 17-30, make the required change in the given equation. \(x^{2}+y^{2}+4 z^{2}=10\) to spherical coordinates

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

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

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

In Problems 17-24, sketch the graphs of the given equations. Begin by sketching the traces in the coordinate planes (see Examples 4 and 5). $$ -3 x+2 y+z=6 $$

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

A wind with velocity 45 miles per hour is blowing in the direction \(\mathrm{N} 20^{\circ} \mathrm{W}\). 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?

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?

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

Find the distance between the skew (nonintersecting and nonparallel) lines \(x=2-t, y=3+4 t, z=2 t\) and \(x=-1+t\), \(y=2, z=-1+2 t\) by using the following steps. (a) Note by setting \(t=0\) that \((2,3,0)\) is on the first line. (b) Find the equation of the plane \(\pi\) through \((2,3,0)\) parallel to both given lines (i.e., with normal perpendicular to both). (c) Find a point \(Q\) on the second line. (d) Find the distance from \(Q\) to the plane \(\pi\). (See Example 10 of Section 11.3.) See Problem 32 for another way to do this problem.

Let \(\mathbf{a}\) and \(\mathbf{b}\) be nonparallel vectors, and let \(\mathbf{c}\) be any nonzero vector. Show that \((\mathbf{a} \times \mathbf{b}) \times \mathbf{c}\) is a vector in the plane of \(\mathbf{a}\) and b.

A ship is sailing due south at 20 miles per hour. A man walks west (i.e., at right angles to the side of the ship) across the deck at 3 miles per hour. What are the magnitude and direction of his velocity relative to the surface of the water?

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

For what values of \(a, b\), and \(c\) are the three vectors \(\langle a, 0,1\rangle,\langle 0,2, b\rangle\), and \(\langle 1, c, 1\rangle\) mutually orthogonal.