Chapter 10

Find the velocity, acceleration, and speed of a particle with the given position function. $$\mathbf{r}(t)=\left\langle t^{2}+t, t^{2}-t, t^{3}\right\rangle$$

Let \(C\) be the curve of intersection of the parabolic cylinder \(x^{2}=2 y\) and the surface \(3 z=x y .\) Find the exact length of \(C\) from the origin to the point \((6,18,36)\) .

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

Describe and sketch the surface. \(x y=1\)

\(6-10=\) Find parametric equations and symmetric equations for the line. The line through the points \(\left(0, \frac{1}{2}, 1\right)\) and \((2,1,-3)\)

Find the cross product a \(\times\) b and verify that it is orthogonal to both a and b. $$\mathbf{a}=\langle t, 1,1 / t\rangle, \quad \mathbf{b}=\left\langle t^{2}, t^{2}, 1\right\rangle$$

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. \(A(0,3,1), \quad B(2,3,-1)\)

2-10 Find \(\mathbf{a} \cdot \mathbf{b}\) $$\mathbf{a}=2 \mathbf{i}+\mathbf{j}, \quad \mathbf{b}=\mathbf{i}-\mathbf{j}+\mathbf{k}$$

Find the lengths of the sides of the triangle \(P Q R .\) Is it a right triangle? Is it an isosceles triangle? (a) \(P(3,-2,-3), \quad Q(7,0,1), \quad R(1,2,1)\) (b) \(P(2,-1,0), \quad Q(4,1,1), \quad R(4,-5,4)\)

Find the velocity, acceleration, and speed of a particle with the given position function. $$\mathbf{r}(t)=\langle 2 \cos t, 3 t, 2 \sin t\rangle$$

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

Describe and sketch the surface. \(z=\sin y\)

\(6-10=\) Find parametric equations and symmetric equations for the line. The line through \((2,1,0)\) and perpendicular to both \(\mathbf{i}+\mathbf{j}\) and \(\mathbf{j}+\mathbf{k}\)

If \(\mathbf{a}=\mathbf{i}-2 \mathbf{k}\) and \(\mathbf{b}=\mathbf{j}+\mathbf{k},\) find a \(\times \mathbf{b} .\) Sketch a, \(\mathbf{b},\) and \(\mathbf{a} \times \mathbf{b}\) as vectors starting at the origin.

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. \(A(4,0,-2), \quad B(4,2,1)\)

Find the distance from \((4,-2,6)\) to each of the following. (a) The \(x y\) -plane (b) The \(y z-\) plane (c) The \(x z\) -plane (d) The \(x\) -axis (e) The \(y\) -axis (f) The \(z\) -axis

2-10 Find \(\mathbf{a} \cdot \mathbf{b}\) $$\mathbf{a}=3 \mathbf{i}+2 \mathbf{j}-\mathbf{k}, \quad \mathbf{b}=4 \mathbf{i}+5 \mathbf{k}$$

Graph the curve with parametric equations \(x=\cos t\) \(y=\sin 3 t, z=\sin t .\) Find the total length of this curve correct to four decimal places.

Find the velocity, acceleration, and speed of a particle with the given position function. $$\mathbf{r}(t)=\sqrt{2} t \mathbf{i}+e^{t} \mathbf{j}+e^{-t} \mathbf{k}$$

Reparametrize the curve with respect to arc length measured from the point where \(t=0\) in the direction of increasing \(t .\) $$\mathbf{r}(t)=2 t \mathbf{i}+(1-3 t) \mathbf{j}+(5+4 t) \mathbf{k}$$

\(5-12=\) Sketch the curve with the given vector equation. Indicate with an arrow the direction in which \(t\) increases. $$ \mathbf{r}(t)=\langle 1, \cos t, 2 \sin t\rangle $$

(a) Find and identify the traces of the quadric surface \(x^{2}+y^{2}-z^{2}=1\) and explain why the graph looks like the graph of the hyperboloid of one sheet in Table \(1 .\) (b) If we change the equation in part (a) to \(x^{2}-y^{2}+z^{2}=1,\) how is the graph affected? (c) What if we change the equation in part (a) to \(x^{2}+y^{2}+2 y-z^{2}=0 ?\)

\(6-10=\) Find parametric equations and symmetric equations for the line. The line through \((1,-1,1)\) and parallel to the line \(x+2=\frac{1}{2} y=z-3\)

Find the vector, not with determinants, but by using properties of cross products $$(\mathbf{i} \times \mathbf{j}) \times \mathbf{k}$$.

Find the sum of the given vectors and illustrate geometrically. $$\langle- 1,4\rangle, \quad\langle 6,-2\rangle$$

2-10 Find \(\mathbf{a} \cdot \mathbf{b}\) $$|\mathbf{a}|=6, \quad|\mathbf{b}|=5, \quad$$ the angle between a and \(\mathbf{b}\) is 2\(\pi / 3\)

Determine whether the points lie on a straight line. (a) \(A(2,4,2), \quad B(3,7,-2), \quad C(1,3,3)\) (b) \(D(0,-5,5), \quad E(1,-2,4), \quad F(3,4,2)\)

Find the velocity, acceleration, and speed of a particle with the given position function. $$\mathbf{r}(t)=t^{2} \mathbf{i}+2 t \mathbf{j}+\ln t \mathbf{k}$$

Reparametrize the curve with respect to arc length measured from the point where \(t=0\) in the direction of increasing \(t .\) $$\mathbf{r}(t)=e^{2 t} \cos 2 t \mathbf{i}+2 \mathbf{j}+e^{2 t} \sin 2 t \mathbf{k}$$

\(5-12=\) Sketch the curve with the given vector equation. Indicate with an arrow the direction in which \(t\) increases. $$ \mathbf{r}(t)=t^{2} \mathbf{i}+t \mathbf{j}+2 \mathbf{k} $$

(a) Find and identify the traces of the quadric surface \(-x^{2}-y^{2}+z^{2}=1\) and explain why the graph looks like the graph of the hyperboloid of two sheets in Table \(1 .\) (b) If the equation in part (a) is changed to \(x^{2}-y^{2}-z^{2}=1,\) what happens to the graph? Sketch the new graph.

\(6-10=\) Find parametric equations and symmetric equations for the line. The line of intersection of the planes \(x+2 y+3 z=1\) and \(x-y+z=1\)

Find the vector, not with determinants, but by using properties of cross products $$\mathbf{k} \times(\mathbf{i}-2 \mathbf{j})$$

Find the sum of the given vectors and illustrate geometrically. $$\langle 3,-1\rangle, \quad\langle- 1,5\rangle$$

2-10 Find \(\mathbf{a} \cdot \mathbf{b}\) $$|\mathbf{a}|=3, \quad|\mathbf{b}|=\sqrt{6}, \quad$$ the angle between a and \(\mathbf{b}\) is \(45^{\circ}\)

Find an equation of the sphere with center \((2,-6,4)\) and radius \(5 .\) Describe its intersection with each of the coordinate planes.

Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and position. $$\mathbf{a}(t)=\mathbf{i}+2 \mathbf{j}, \quad \mathbf{v}(0)=\mathbf{k}, \quad \mathbf{r}(0)=\mathbf{i} $$

Suppose you start at the point \((0,0,3)\) and move 5 units along the curve \(x=3 \sin t, y=4 t, z=3 \cos t\) in the positive direction. Where are you now?

\(5-12=\) Sketch the curve with the given vector equation. Indicate with an arrow the direction in which \(t\) increases. $$ \mathbf{r}(t)=t^{2} \mathbf{i}+t^{4} \mathbf{j}+t^{6} \mathbf{k} $$

Use traces to sketch and identify the surface. \(x=y^{2}+4 z^{2}\)

Is the line through \((-4,-6,1)\) and \((-2,0,-3)\) parallel to the line through \((10,18,4)\) and \((5,3,14) ?\)

Find the vector, not with determinants, but by using properties of cross products $$(\mathbf{j}-\mathbf{k}) \times(\mathbf{k}-\mathbf{i})$$

Find the sum of the given vectors and illustrate geometrically. $$\langle 3,0,1\rangle, \quad\langle 0,8,0\rangle$$

Find an equation of the sphere that passes through the point \((4,3,-1)\) and has center \((3,8,1) .\)

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which increases. $$\mathbf{r}(t)=\cos t \mathbf{i}-\cos t \mathbf{j}+\sin t \mathbf{k}$$

Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and position. $$\mathbf{a}(t)=2 \mathbf{i}+6 t \mathbf{j}+12 t^{2} \mathbf{k}, \quad \mathbf{v}(0)=\mathbf{i}, \quad \mathbf{r}(0)=\mathbf{j}-\mathbf{k}$$

Reparametrize the curve $$\mathbf{r}(t)=\left(\frac{2}{t^{2}+1}-1\right) \mathbf{i}+\frac{2 t}{t^{2}+1} \mathbf{j}$$ with respect to arc length measured from the point \((1,0)\) in the direction of increasing \(t\) . Express the reparametrization in its simplest form. What can you conclude about the curve?

Use traces to sketch and identify the surface. \(9 x^{2}-y^{2}+z^{2}=0\)

Is the line through \((-2,4,0)\) and \((1,1,1)\) perpendicular to the line through \((2,3,4)\) and \((3,-1,-8) ?\)

Find the vector, not with determinants, but by using properties of cross products $$(\mathbf{i}+\mathbf{j}) \times(\mathbf{i}-\mathbf{j})$$