Chapter 11

Name and sketch the graph of each of the following equations in three-space. $$ 9 x^{2}-y^{2}+9 z^{2}-9=0 $$

find the unit tangent vector \(\mathbf{T}(t)\) and the curvature \(\kappa(t)\) at the point where \(t=t_{1} .\) For calculating \(\kappa\), we suggest using Theorem \(A\), as in Example \(5 .\) $$ \mathbf{r}(t)=\frac{1}{3} t^{3} \mathbf{i}+\frac{1}{2} t^{2} \mathbf{j} ; t_{1}=1 $$

Write both the parametric equations and the symmetric equations for the line through the given point parallel to the given vector. \((-2,2,-2),(7,-6,3)\)

Find the required limit or indicate that it does not exist. $$ \lim _{t \rightarrow 0^{-}}\left\langle e^{-1 / t^{2}}, \frac{t}{|t|},|t|\right\rangle $$

Find the area of the parallelogram with \(\mathbf{a}=2 \mathbf{i}+2 \mathbf{j}-\mathbf{k}\) and \(\mathbf{b}=-\mathbf{i}+\mathbf{j}-4 \mathbf{k}\) as the adjacent sides.

Let \(\quad \mathbf{a}=\langle\sqrt{3} / 3, \sqrt{3} / 3, \sqrt{3} / 3\rangle, \mathbf{b}=\langle 1,-1,0\rangle, \quad\) and \(\mathbf{c}=\langle-2,-2,1\rangle .\) Find the angle between each pair of vectors.

Find the distance from \((2,3,-1)\) to (a) the \(x y\) -plane, (b) the \(y\) -axis, and (c) the origin.

Name and sketch the graph of each of the following equations in three-space. $$ 4 x^{2}+16 y^{2}-32 z=0 $$

Sketch the graph of the given cylindrical or spherical equation. $$ \phi=\pi / 6 $$

Find the symmetric equations of the line of intersection of the given pair of planes. \(4 x+3 y-7 z=1,10 x+6 y-5 z=10\)

When no domain is given in the definition of a vectorvalued function, it is to be understood that the domain is the set of all (real) scalars for which the rule for the function makes sense and gives real vectors (i.e., vectors with real components). Find the domain of each of the following vector-valued functions: (a) \(\mathbf{r}(t)=\frac{2}{t-4} \mathbf{i}+\sqrt{3-t} \mathbf{j}+\ln |4-t| \mathbf{k}\) (b) \(\mathbf{r}(t)=\left[t^{2}\right] \mathbf{i}-\sqrt{20-t} \mathbf{j}+3 \mathbf{k}([]\) denotes the greatest integer function.) (c) \(\mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+\sqrt{9-t^{2}} \mathbf{k}\)

A rectangular box has its faces parallel to the coordinate planes and has \((2,3,4)\) and \((6,-1,0)\) as the end points of a main diagonal. Sketch the box and find the coordinates of all eight vertices.

Name and sketch the graph of each of the following equations in three-space. $$ -x^{2}+y^{2}+z^{2}=0 $$

Sketch the graph of the given cylindrical or spherical equation. $$ \theta=\pi / 6 $$

Find the symmetric equations of the line of intersection of the given pair of planes. \(x+y-z=2,3 x-2 y+z=3\)

State the domain of each of the following vector-valued functions: (a) \(\mathbf{r}(t)=\ln (t-1) \mathbf{i}+\sqrt{20-t} \mathbf{j}\) (b) \(\mathbf{r}(t)=\ln \left(t^{-1}\right) \mathbf{i}+\tan ^{-1} t \mathbf{j}+t \mathbf{k}\) (c) \(\mathbf{r}(t)=\frac{1}{\sqrt{1-t^{2}}} \mathbf{j}+\frac{1}{\sqrt{9-t^{2}}} \mathbf{k}\)

Find the sum \(\mathbf{u}+\mathbf{v}\), the difference \(\mathbf{u}-\mathbf{v}\), and the magnitudes \(\|\mathbf{u}\|\) and \(\|\mathbf{v}\|\) $$ \mathbf{u}=\langle 0,0\rangle, \mathbf{v}=\langle-3,4\rangle $$

\(P(x, 5, z)\) is on a line through \(Q(2,-4,3)\) that is parallel to one of the coordinate axes. Which axis must it be and what are \(x\) and \(z ?\)

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

Sketch the graph of the given cylindrical or spherical equation. $$ r=3 \cos \theta $$

find the unit tangent vector \(\mathbf{T}(t)\) and the curvature \(\kappa(t)\) at the point where \(t=t_{1} .\) For calculating \(\kappa\), we suggest using Theorem \(A\), as in Example \(5 .\) $$ x(t)=1-t^{2}, y(t)=1-t^{3} ; t_{1}=1 $$

Find the symmetric equations of the line of intersection of the given pair of planes. \(x+4 y-2 z=13,2 x-y-2 z=5\)

Find the equation of the plane through the given points. $$ (1,3,2),(0,3,0), \text { and }(2,4,3) $$

Show that the vectors \(\langle 6,3\rangle\) and \(\langle-1,2\rangle\) are orthogonal.

Find the sum \(\mathbf{u}+\mathbf{v}\), the difference \(\mathbf{u}-\mathbf{v}\), and the magnitudes \(\|\mathbf{u}\|\) and \(\|\mathbf{v}\|\) $$ \mathbf{u}=\langle 12,12\rangle, \mathbf{v}=\langle-2,2\rangle $$

Write the equation of the sphere with the given center and radius. (a) \((1,2,3) ; 5\) (b) \((-2,-3,-6) ; \sqrt{5}\) (c) \((\pi, e, \sqrt{2}) ; \sqrt{\pi}\)

Sketch the graph of the given cylindrical or spherical equation. $$ r=2 \sin 2 \theta $$

Find the symmetric equations of the line of intersection of the given pair of planes. \(x-3 y+z=-1,6 x-5 y+4 z=9\)

Find the equation of the plane through the given points. $$ (1,1,2),(0,0,1), \text { and }(-2,-3,0) $$

Show that the vectors \(\mathbf{a}=\langle 1,1,1\rangle, \mathbf{b}=\langle 1,-1,0\rangle\), and \(\mathbf{c}=\langle-1,-1,2\rangle\) are mutually orthogonal, that is, each pair of vectors is orthogonal.

Find the sum \(\mathbf{u}+\mathbf{v}\), the difference \(\mathbf{u}-\mathbf{v}\), and the magnitudes \(\|\mathbf{u}\|\) and \(\|\mathbf{v}\|\) $$ \mathbf{u}=\langle-0.2,0.8\rangle, \mathbf{v}=\langle-2.1,1.3\rangle $$

Find the equation of the sphere whose center is \((2,4,5)\) and that is tangent to the \(x y\) -plane.

Name and sketch the graph of each of the following equations in three-space. $$ x^{2}-z^{2}+y=0 $$

Sketch the graph of the given cylindrical or spherical equation. $$ \rho=3 \cos \phi $$

Find \(D_{t} \mathbf{r}(t)\) and \(D_{t}^{2} \mathbf{r}(t)\) for each of the following: (a) \(\mathbf{r}(t)=(3 t+4)^{3} \mathbf{i}+e^{i^{2}} \mathbf{j}+\mathbf{k}\) (b) \(\mathbf{r}(t)=\sin ^{2} t \mathbf{i}+\cos 3 t \mathbf{j}+t^{2} \mathbf{k}\)

Find the symmetric equations of the line through \((4,0,6)\) and perpendicular to the plane \(x-5 y+2 z=10\).

Find the equation of the plane through the given points. $$ (7,0,0),(0,3,0), \text { and }(0,0,5) $$

Show that the vectors \(\mathbf{a}=\mathbf{i}-\mathbf{j}, \mathbf{b}=\mathbf{i}+\mathbf{j}, \quad\) and \(\mathbf{c}=2 \mathbf{k}\) are mutually orthogonal, that is, each pair of vectors is orthogonal.

Find the sum \(\mathbf{u}+\mathbf{v}\), the difference \(\mathbf{u}-\mathbf{v}\), and the magnitudes \(\|\mathbf{u}\|\) and \(\|\mathbf{v}\|\) $$ \mathbf{u}=\langle-1,0,0\rangle, \mathbf{v}=\langle 3,4,0\rangle $$

Name and sketch the graph of each of the following equations in three-space. $$ x^{2}+y^{2}-4 z^{2}+4=0 $$

Sketch the graph of the given cylindrical or spherical equation. $$ \rho=\sec \phi $$

Find \(\mathbf{r}^{\prime}(t)\) and \(\mathbf{r}^{\prime \prime}(t)\) for each of the following: (a) \(\mathbf{r}(t)=\left(e^{t}+e^{-t^{2}}\right) \mathbf{i}+2^{t} \mathbf{j}+t \mathbf{k}\) (b) \(\mathbf{r}(t)=\tan 2 t \mathbf{i}+\arctan t \mathbf{j}\)

Find the symmetric equations of the line through \((-5,7,-2)\) and perpendicular to both \(\langle 2,1,-3\rangle\) and \(\langle 5,4,-1\rangle\).

Find the equation of the plane through the given points. $$ (a, 0,0),(0, b, 0) \text { , and }(0,0, c), \text { (None of } a, b \text { , and } c \text { is zero.) } $$

If \(\mathbf{u}+\mathbf{v}\) is orthogonal to \(\mathbf{u}-\mathbf{v}\), what can you say about the relative magnitudes of \(\mathbf{u}\) and \(\mathbf{v}\) ?

Find the sum \(\mathbf{u}+\mathbf{v}\), the difference \(\mathbf{u}-\mathbf{v}\), and the magnitudes \(\|\mathbf{u}\|\) and \(\|\mathbf{v}\|\) $$ \mathbf{u}=\langle 0,0,0\rangle, \mathbf{v}=\langle-3,3,1\rangle $$

Complete the squares to find the center and radius of the sphere whose equation is given (see Example 2). \(x^{2}+y^{2}+z^{2}+2 x-6 y-10 z+34=0\)

Sketch the graph of the given cylindrical or spherical equation. $$ r^{2}+z^{2}=9 $$

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

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