Chapter 10

Find the velocity, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and draw the velocity and acceleration vectors for the specified value of \(t .\) $$\mathbf{r}(t)=\left\langle-\frac{1}{2} t^{2}, t\right\rangle, \quad t=2$$

Find the length of the curve. $$\mathbf{r}(t)=\langle t, 3 \cos t, 3 \sin t\rangle, \quad-5 \leqslant t \leqslant 5$$

\(1-2 \approx\) Find the domain of the vector function. $$\mathbf{r}(t)=\left\langle\sqrt{4-t^{2}}, e^{-3 t}, \ln (t+1)\right\rangle$$

(a) What does the equation \(y=x^{2}\) represent as a curve in \(\mathbb{R}^{2} ?\) (b) What does it represent as a surface in \(\mathbb{R}^{3} ?\) (c) What does the equation \(z=y^{2}\) represent?

Determine whether each statement is true or false. (a) Two lines parallel to a third line are parallel. (b) Two lines perpendicular to a third line are parallel. (c) Two planes parallel to a third plane are parallel. (d) Two planes perpendicular to a third plane are parallel. (e) Two lines parallel to a plane are parallel. (f) Two lines perpendicular to a plane are parallel. (g) Two planes parallel to a line are parallel. (h) Two planes perpendicular to a line are parallel. (i) Two planes either intersect or are parallel. (j) Two lines either intersect or are parallel. (k) A plane and a line either intersect or are parallel.

Find the cross product a \(\times\) b and verify that it is orthogonal to both a and b. $$\mathbf{a}=\langle 6,0,-2\rangle, \quad \mathbf{b}=\langle 0,8,0\rangle$$

Which of the following expressions are meaningful? Which are meaningless? Explain. $$ \begin{array}{ll}{\text { (a) }(\mathbf{a} \cdot \mathbf{b}) \cdot \mathbf{c}} & {\text { (b) }(\mathbf{a} \cdot \mathbf{b}) \mathbf{c}} \\\ {(\mathrm{c})|\mathbf{a}|(\mathbf{b} \cdot \mathbf{c})} & {\text { (d) } \mathbf{a} \cdot(\mathbf{b}+\mathbf{c})} \\ {\text { (e) } \mathbf{a} \cdot \mathbf{b}+\mathbf{c}} & {\text { (f) }|\mathbf{a}| \cdot(\mathbf{b}+\mathbf{c})}\end{array} $$

Suppose you start at the origin, move along the \(x\) -axis a distance of 4 units in the positive direction, and then move downward a distance of 3 units. What are the coordinates of your position?

Find the velocity, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and draw the velocity and acceleration vectors for the specified value of \(t .\) $$\mathbf{r}(t)=\langle 2-t, 4 \sqrt{t}\rangle, \quad t=1$$

Find the length of the curve. $$\mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+\ln \cos t \mathbf{k}, \quad 0 \leqslant t \leqslant \pi / 4$$

\(1-2 \approx\) Find the domain of the vector function. $$\mathbf{r}(t)=\frac{t-2}{t+2} \mathbf{i}+\sin t \mathbf{j}+\ln \left(9-t^{2}\right) \mathbf{k}$$

(a) Sketch the graph of \(y=e^{x}\) as a curve in \(\mathbb{R}^{2}\) . (b) Sketch the graph of \(y=e^{x}\) as a surface in \(\mathbb{R}^{3}\) . (c) Describe and sketch the surface \(z=e^{y}\)

\(2-5=\) Find a vector equation and parametric equations for the line. The line through the point \((6,-5,2)\) and parallel to the vector \(\left\langle 1,3,-\frac{2}{3}\right\rangle\)

Find the cross product a \(\times\) b and verify that it is orthogonal to both a and b. $$\mathbf{a}=\langle 1,1,-1\rangle, \quad \mathbf{b}=\langle 2,4,6\rangle$$

2-10 Find \(\mathbf{a} \cdot \mathbf{b}\) $$\mathbf{a}=\langle- 2,3\rangle, \quad \mathbf{b}=\langle 0.7,1.2\rangle$$

Sketch the points \((0,5,2),(4,0,-1),(2,4,6),\) and \((1,-1,2)\) on a single set of coordinate axes.

Find the velocity, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and draw the velocity and acceleration vectors for the specified value of \(t .\) $$\mathbf{r}(t)=3 \cos t \mathbf{i}+2 \sin t \mathbf{j}, \quad t=\pi / 3$$

Find the length of the curve. $$\mathbf{r}(t)=\mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k}, \quad 0 \leq t \leqslant 1$$

Describe and sketch the surface. \(x^{2}+z^{2}=1\)

\(2-5=\) Find a vector equation and parametric equations for the line. The line through the point \((2,2.4,3.5)\) and parallel to the vector \(3 \mathbf{i}+2 \mathbf{j}-\mathbf{k}\)

Find the cross product a \(\times\) b and verify that it is orthogonal to both a and b. $$\mathbf{a}=\mathbf{i}+3 \mathbf{j}-2 \mathbf{k}, \quad \mathbf{b}=-\mathbf{i}+5 \mathbf{k}$$

2-10 Find \(\mathbf{a} \cdot \mathbf{b}\) $$\mathbf{a}=\left\langle- 2, \frac{1}{3}\right\rangle, \quad \mathbf{b}=\langle- 5,12\rangle$$

Which of the points \(A(-4,0,-1), B(3,1,-5),\) and \(C(2,4,6)\) is closest to the \(y z\) -plane? Which point lies in the \(x z\) -plane?

\(3-4=\) Find the limit $$\lim _{t \rightarrow 0}\left(e^{-3 / \mathbf{i}} \mathbf{i}+\frac{t^{2}}{\sin ^{2} t} \mathbf{j}+\cos 2 t \mathbf{k}\right)$$

Find the velocity, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and draw the velocity and acceleration vectors for the specified value of \(t .\) $$\mathbf{r}(t)=e^{t} \mathbf{i}+e^{2 t} \mathbf{j}, \quad t=0$$

\(3-4=\) Find the limit $$\lim _{t \rightarrow 1}\left(\frac{t^{2}-t}{t-1} \mathbf{i}+\sqrt{t+8} \mathbf{j}+\frac{\sin \pi t}{\ln t} \mathbf{k}\right)$$

Find the length of the curve. $$\mathbf{r}(t)=12 t \mathbf{i}+8 t^{3 / 2} \mathbf{j}+3 t^{2} \mathbf{k}, \quad 0 \leqslant t \leqslant 1$$

Describe and sketch the surface. \(4 x^{2}+y^{2}=4\)

\(2-5=\) Find a vector equation and parametric equations for the line. The line through the point \((0,14,-10)\) and parallel to the line \(x=-1+2 t, y=6-3 t, z=3+9 t\)

Find the cross product a \(\times\) b and verify that it is orthogonal to both a and b. $$\mathbf{a}=\mathbf{j}+7 \mathbf{k}, \quad \mathbf{b}=2 \mathbf{i}-\mathbf{j}+4 \mathbf{k}$$

2-10 Find \(\mathbf{a} \cdot \mathbf{b}\) $$\mathbf{a}=\langle 6,-2,3\rangle, \quad \mathbf{b}=\langle 2,5,-1\rangle$$

What are the projections of the point \((2,3,5)\) on the \(x y-y z-\) and \(x z-\) planes? Draw a rectangular box with the origin and \((2,3,5)\) as opposite vertices and with its faces parallel to the coordinate planes. Label all vertices of the box. Find the length of the diagonal of the box.

Find the velocity, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and draw the velocity and acceleration vectors for the specified value of \(t .\) $$\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+2 \mathbf{k}, \quad t=1$$

Find the length of the curve correct to four decimal places, (Use your calculator to approximate the integral.) $$\mathbf{r}(t)=\left\langle t^{2}, t^{3}, t^{4}\right\rangle, \quad 0 \leq t \leqslant 2$$

\(5-12=\) Sketch the curve with the given vector equation. Indicate with an arrow the direction in which \(t\) increases. $$ \mathbf{r}(t)=\langle\sin t, t\rangle $$

Describe and sketch the surface. \(z=1-y^{2}\)

\(2-5=\) Find a vector equation and parametric equations for the line. The line through the point \((1,0,6)\) and perpendicular to the plane \(x+3 y+z=5\)

Find the cross product a \(\times\) b and verify that it is orthogonal to both a and b. $$\mathbf{a}=\mathbf{i}-\mathbf{j}-\mathbf{k}, \quad \mathbf{b}=\frac{1}{2} \mathbf{i}+\mathbf{j}+\frac{1}{2} \mathbf{k}$$

Find a vector a with representation given by the directed line segment \(\overline{A B} .\) Draw \(\vec{A B}\) and the equivalent representation starting at the origin. \(A(-1,1), \quad B(3,2)\)

2-10 Find \(\mathbf{a} \cdot \mathbf{b}\) $$\mathbf{a}=\left\langle 4,1, \frac{1}{4}\right\rangle, \quad \mathbf{b}=\langle 6,-3,-8\rangle$$

Describe and sketch the surface in \(\mathbb{R}^{3}\) represented by the equation \(x+y=2\)

Find the velocity, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and draw the velocity and acceleration vectors for the specified value of \(t .\) $$\mathbf{r}(t)=t \mathbf{i}+2 \cos t \mathbf{j}+\sin t \mathbf{k}, \quad t=0$$

Find the length of the curve correct to four decimal places, (Use your calculator to approximate the integral.) $$\mathbf{r}(t)=\left\langle t, e^{-t}, t e^{-t}\right\rangle, \quad 1 \leq t \leqslant 3$$

\(5-12=\) Sketch the curve with the given vector equation. Indicate with an arrow the direction in which \(t\) increases. $$ \mathbf{r}(t)=\left\langle t^{3}, t^{2}\right\rangle $$

\(6-10\) " Find parametric equations and symmetric equations for the line. The line through the points \((1.0,2.4,4.6)\) and \((2.6,1.2,0.3)\)

Find the cross product a \(\times\) b and verify that it is orthogonal to both a and b. $$\mathbf{a}=t \mathbf{i}+\cos t \mathbf{j}+\sin t \mathbf{k}, \quad \mathbf{b}=\mathbf{i}-\sin t \mathbf{j}+\cos t \mathbf{k}$$

Find a vector a with representation given by the directed line segment \(\overline{A B} .\) Draw \(\vec{A B}\) and the equivalent representation starting at the origin. \(A(-4,-1), \quad B(1,2)\)

2-10 Find \(\mathbf{a} \cdot \mathbf{b}\) $$\mathbf{a}=\langle p,-p, 2 p\rangle, \quad \mathbf{b}=\langle 2 q, q,-q\rangle$$

(a) What does the equation \(x=4\) represent in \(\mathbb{R}^{2} ?\) What does it represent in \(\mathbb{R}^{3}\) ? Illustrate with sketches. (b) What does the equation \(y=3\) represent in \(\mathbb{R}^{3} ?\) What does \(z=5\) represent? What does the pair of equations \(y=3, z=5\) represent? In other words, describe the set of points \((x, y, z)\) such that \(y=3\) and \(z=5 .\) Illustrate with a sketch.

Describe and sketch the surface. \(y=z^{2}\)