Chapter 11

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.

In Problems 7-16, sketch the graph of the given cylindrical or spherical equation. \(\rho=5\)

$$ \lim _{t \rightarrow 0^{-}}\left\langle e^{-1 / t^{2}}, \frac{t}{|t|},|t|\right\rangle $$

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

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

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 area of the triangle with \((3,2,1),(2,4,6)\), and \((-1,2,5)\) as vertices.

For the two-dimensional vectors \(\mathbf{u}\) and \(\mathbf{v}\) in Problems \(9-12\), find the sum \(\mathbf{u}+\mathbf{v}\), the difference \(\mathbf{u}-\mathbf{v}\), and the magnitudes \(\|\mathbf{u}\|\) and \(\|\mathbf{v}\|\). $$ \mathbf{u}=\langle-1,0\rangle, \mathbf{v}=\langle 3,4\rangle $$

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) \(\mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+\sqrt{9-t^{2}} \mathbf{k}\)

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

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.

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

For the two-dimensional vectors \(\mathbf{u}\) and \(\mathbf{v}\) in Problems \(9-12\), find the sum \(\mathbf{u}+\mathbf{v}\), the difference \(\mathbf{u}-\mathbf{v}\), and the magnitudes \(\|\mathbf{u}\|\) and \(\|\mathbf{v}\|\). $$ \mathbf{u}=\langle 0,0\rangle, \mathbf{v}=\langle-3,4\rangle $$

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 \(5 .\) \(\mathbf{r}(t)=e^{t} \mathbf{i}+e^{t} \mathbf{j} ; t_{1}=\ln 2\)

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) \(\mathbf{r}(t)=\frac{1}{\sqrt{1-t^{2}}} \mathbf{j}+\frac{1}{\sqrt{9-t^{2}}} \mathbf{k}\)

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

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

In Problems \(11-14\), find the equation of the plane through the given points. \((1,3,2),(0,3,0)\), and \((2,4,3)\)

For the two-dimensional vectors \(\mathbf{u}\) and \(\mathbf{v}\) in Problems \(9-12\), find the sum \(\mathbf{u}+\mathbf{v}\), the difference \(\mathbf{u}-\mathbf{v}\), and the magnitudes \(\|\mathbf{u}\|\) and \(\|\mathbf{v}\|\). $$ \mathbf{u}=\langle 12,12\rangle, \mathbf{v}=\langle-2,2\rangle $$

In Problems 7-16, sketch the graph of the given cylindrical or spherical equation. \(r=3 \cos \theta\)

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

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

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

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

In Problems \(11-14\), find the equation of the plane through the given points. \((1,1,2),(0,0,1)\), and \((-2,-3,0)\)

For the two-dimensional vectors \(\mathbf{u}\) and \(\mathbf{v}\) in Problems \(9-12\), find the sum \(\mathbf{u}+\mathbf{v}\), the difference \(\mathbf{u}-\mathbf{v}\), and the magnitudes \(\|\mathbf{u}\|\) and \(\|\mathbf{v}\|\). $$ \mathbf{u}=\langle-0.2,0.8\rangle, \mathbf{v}=\langle-2.1,1.3\rangle $$

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 equation of the sphere whose center is \((2,4,5)\) and that is tangent to the \(x y\)-plane. In Problems 13-16, complete the squares to find the center and radius of the sphere whose equation is given (see Example 2).

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

In Problems \(11-14\), find the equation of the plane through the given points. \((7,0,0),(0,3,0)\), and \((0,0,5)\)

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 \(\|\mathbf{v}\|\). $$ \mathbf{u}=\langle-1,0,0\rangle, \mathbf{v}=\langle 3,4,0\rangle $$

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

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

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

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}-12 x+14 y-8 z+1=0

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

In Problems \(11-14\), find the equation of the plane through the given points. \((a, 0,0),(0, b, 0)\), and \((0,0, c),(\) None of \(a, b\), and \(c\) is zero. \()\)

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 \(\|\mathbf{v}\|\). $$ \mathbf{u}=\langle 0,0,0\rangle, \mathbf{v}=\langle-3,3,1\rangle $$

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) \(\mathbf{r}(t)=\tan 2 t \mathbf{i}+\arctan t \mathbf{j}\)

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

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

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

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 \(\|\mathbf{v}\|\). $$ \mathbf{u}=\langle 1,0,1\rangle, \mathbf{v}=\langle-5,0,0\rangle $$

In Problems 7-16, 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, \(y=2 x^{2},(1,2)\)

$$ \text { If } \mathbf{r}(t)=e^{-t} \mathbf{i}-\ln \left(t^{2}\right) \mathbf{j} \text {, find } D_{t}\left[\mathbf{r}(t) \cdot \mathbf{r}^{\prime \prime}(t)\right] \text {. } $$

Find two vectors of length 10 , each of which is perpendicular to both \(-4 \mathbf{i}+5 \mathbf{j}+\mathbf{k}\) and \(4 \mathbf{i}+\mathbf{j}\).

Name and sketch the graph of each of the following equations in three-space. $$ 9 x^{2}+4 z^{2}-36 y=0 $$