Chapter 10

Find the sum of the given vectors and illustrate geometrically. $$\langle 1,3,-2\rangle, \quad\langle 0,0,6\rangle$$

Find an equation of the sphere that passes through the origin and whose center is \((1,2,3) .\)

(a) Find the position vector of a particle that has the given acceleration and the specified initial velocity and position. (b) Use a computer to graph the path of the particle. $$\mathbf{a}(t)=2 t \mathbf{i}+\sin t \mathbf{j}+\cos 2 t \mathbf{k}, \quad \mathbf{v}(0)=\mathbf{i}, \quad \mathbf{r}(0)=\mathbf{j}$$

(a) Find the unit tangent and unit normal vectors \(\mathbf{T}(t)\) and \(\mathbf{N}(t)\) . (b) Use Formula 9 to find the curvature. $$\mathbf{r}(t)=\langle t, 3 \cos t, 3 \sin t\rangle$$

\(13-16=\) Find a vector equation and parametric equations for the line segment that joins \(P\) to \(Q .\) $$P(2,0,0), \quad Q(6,2,-2)$$

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

(a) Find symmetric equations for the line that passes through the point \((1,-5,6)\) and is parallel to the vector \(\langle- 1,2,-3\rangle .\) (b) Find the points in which the required line in part (a) intersects the coordinate planes.

State whether each expression is meaningful. If not, explain why. If so, state whether it is a vector or a scalar. $$\begin{array}{ll}{\text { (a) } \mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})} & {\text { (b) } \mathbf{a} \times(\mathbf{b} \cdot \mathbf{c})} \\\ {(\mathbf{c}) \mathbf{a} \times(\mathbf{b} \times \mathbf{c})} & {\text { (d) } \mathbf{a} \cdot(\mathbf{b} \cdot \mathbf{c})} \\ { (e) } {(\mathbf{a} \cdot \mathbf{b}) \times(\mathbf{c} \cdot \mathbf{d})} & {\text { (f) }(\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times \mathbf{d})}\end{array}$$

Find \(\mathbf{a}+\mathbf{b}, 2 \mathbf{a}+3 \mathbf{b},|\mathbf{a}|,\) and \(|\mathbf{a}-\mathbf{b}|\) \(\mathbf{a}=\langle 5,-12\rangle, \quad \mathbf{b}=\langle- 3,-6\rangle\)

Show that the equation represents a sphere, and find its center and radius. \(x^{2}+y^{2}+z^{2}-2 x-4 y+8 z=15\)

(a) Show that \(\mathbf{i} \cdot \mathbf{j}=\mathbf{j} \cdot \mathbf{k}=\mathbf{k} \cdot \mathbf{i}=0\) (b) Show that \(\mathbf{i} \cdot \mathbf{i}=\mathbf{j} \cdot \mathbf{j}=\mathbf{k} \cdot \mathbf{k}=1\)

(a) Find the unit tangent and unit normal vectors \(\mathbf{T}(t)\) and \(\mathbf{N}(t)\) . (b) Use Formula 9 to find the curvature. $$\mathbf{r}(t)=\left\langle t^{2}, \sin t-t \cos t, \cos t+t \sin t\right\rangle, \quad t>0$$

\(13-16=\) Find a vector equation and parametric equations for the line segment that joins \(P\) to \(Q .\) $$P(-1,2,-2), \quad Q(-3,5,1)$$

Use traces to sketch and identify the surface. \(25 x^{2}+4 y^{2}+z^{2}=100\)

(a) Find parametric equations for the line through \((2,4,6)\) that is perpendicular to the plane \(x-y+3 z=7\) (b) In what points does this line intersect the coordinate planes?

Find \(\mathbf{a}+\mathbf{b}, 2 \mathbf{a}+3 \mathbf{b},|\mathbf{a}|,\) and \(|\mathbf{a}-\mathbf{b}|\) \(\mathbf{a}=4 \mathbf{i}+\mathbf{j}, \quad \mathbf{b}=\mathbf{i}-2 \mathbf{j}\)

Show that the equation represents a sphere, and find its center and radius. \(x^{2}+y^{2}+z^{2}+8 x-6 y+2 z+17=0\)

A street vendor sells \(a\) hamburgers, \(b\) hot dogs, and \(c\) soft drinks on a given day. He charges \(\$ 2\) for a hamburger, \(\$ 1.50\) for a hot dog, and \(\$ 1\) for a soft drink. If \(\mathbf{A}=\langle a, b, c\rangle\) and \(\mathbf{P}=\langle 2,1.5,1\rangle,\) what is the meaning of the dot product \(\mathbf{A} \cdot \mathbf{P} ?\)

The position function of a particle is given by \(\mathbf{r}(t)=\left\langle t^{2}, 5 t, t^{2}-16 t\right\rangle .\) When is the speed a minimum?

(a) Find the unit tangent and unit normal vectors \(\mathbf{T}(t)\) and \(\mathbf{N}(t)\) . (b) Use Formula 9 to find the curvature. $$\mathbf{r}(t)=\left\langle\sqrt{2} t, e^{t}, e^{-t}\right\rangle$$

\(13-16=\) Find a vector equation and parametric equations for the line segment that joins \(P\) to \(Q .\) $$P(0,-1,1), \quad Q\left(\frac{1}{2}, \frac{1}{3}, \frac{1}{4}\right)$$

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

Find parametric equations for the line segment from \((10,3,1)\) to \((5,6,-3) .\)

Find \(\mathbf{a}+\mathbf{b}, 2 \mathbf{a}+3 \mathbf{b},|\mathbf{a}|,\) and \(|\mathbf{a}-\mathbf{b}|\) \(\mathbf{a}=\mathbf{i}+2 \mathbf{j}-3 \mathbf{k}, \quad \mathbf{b}=-2 \mathbf{i}-\mathbf{j}+5 \mathbf{k}\)

Show that the equation represents a sphere, and find its center and radius. \(2 x^{2}+2 y^{2}+2 z^{2}=8 x-24 z+1\)

\(15-17\) Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) $$\mathbf{a}=\langle 4,3\rangle, \quad \mathbf{b}=\langle 2,-1\rangle$$

What force is required so that a particle of mass \(m\) has the position function \(\mathbf{r}(t)=t^{3} \mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k} ?\)

Use traces to sketch and identify the surface. \(4 x^{2}+9 y^{2}+z=0\)

Find a vector equation for the line segment from \((2,-1,4)\) to \((4,6,1) .\)

Find \(\mathbf{a}+\mathbf{b}, 2 \mathbf{a}+3 \mathbf{b},|\mathbf{a}|,\) and \(|\mathbf{a}-\mathbf{b}|\) \(\mathbf{a}=2 \mathbf{i}-4 \mathbf{j}+4 \mathbf{k}, \quad \mathbf{b}=2 \mathbf{j}-\mathbf{k}\)

Show that the equation represents a sphere, and find its center and radius. \(3 x^{2}+3 y^{2}+3 z^{2}=10+6 y+12 z\)

\(15-17\) Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) $$\mathbf{a}=\langle 4,0,2\rangle, \quad \mathbf{b}=\langle 2,-1,0\rangle$$

(a) Find the unit tangent and unit normal vectors \(\mathbf{T}(t)\) and \(\mathbf{N}(t)\) . (b) Use Formula 9 to find the curvature. $$\mathbf{r}(t)=\left\langle t, \frac{1}{2} t^{2}, t^{2}\right\rangle$$

Use traces to sketch and identify the surface. \(36 x^{2}+y^{2}+36 z^{2}=36\)

If $$\mathbf{a}=\langle 2,-1,3\rangle \text { and } \mathbf{b}=\langle 4,2,1\rangle, \text { find a } \times \mathbf{b} \text { and } \mathbf{b} \times \mathbf{a}.$$

\(17-20=\) Determine whether the lines \(L_{1}\) and \(L_{2}\) are parallel, skew, or intersecting. If they intersect, find the point of intersection. $$ \begin{array}{l}{L_{1} : x=3+2 t, \quad y=4-t, \quad z=1+3 t} \\ {L_{2} : x=1+4 s, \quad y=3-2 s, \quad z=4+5 s}\end{array} $$

(a) Prove that the midpoint of the line segment from \(P_{1}\left(x_{1}, y_{1}, z_{1}\right)\) to \(P_{2}\left(x_{2}, y_{2}, z_{2}\right)\) is $$\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}, \frac{z_{1}+z_{2}}{2}\right)$$ (b) Find the lengths of the medians of the triangle with vertices \(A(1,2,3), B(-2,0,5),\) and \(C(4,1,5) .\)

Find a unit vector with the same direction as \(8 \mathbf{i}-\mathbf{j}+4 \mathbf{k}\)

\(15-17\) Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) $$\mathbf{a}=4 \mathbf{i}-3 \mathbf{j}+\mathbf{k}, \quad \mathbf{b}=2 \mathbf{i}-\mathbf{k}$$

A force with magnitude 20 \(\mathrm{N}\) acts directly upward from the \(x y\) -plane on an object with mass 4 \(\mathrm{kg}\) . The object starts at the origin with initial velocity \(\mathbf{v}(0)=\mathbf{i}-\mathbf{j} .\) Find its position function and its speed at time \(t\)

Show that if a particle moves with constant speed, then the velocity and acceleration vectors are orthogonal.

Use traces to sketch and identify the surface. \(4 x^{2}-16 y^{2}+z^{2}=16\)

\(\begin{array}{l}{\text { If } \mathbf{a}=\langle 1,0,1\rangle, \mathbf{b}=\langle 2,1,-1\rangle, \text { and } \mathbf{c}=\langle 0,1,3\rangle, \text { show }} \\ {\text { that } \mathbf{a} \times(\mathbf{b} \times \mathbf{c}) \neq(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}}\end{array}\)

\(17-20=\) Determine whether the lines \(L_{1}\) and \(L_{2}\) are parallel, skew, or intersecting. If they intersect, find the point of intersection. $$ \begin{array}{l}{L_{1 :} x=5-12 t, \quad y=3+9 t, \quad z=1-3 t} \\ {L_{2} : x=3+8 s, \quad y=-6 s, \quad z=7+2 s}\end{array} $$

Find an equation of a sphere if one of its diameters has end-points \((2,1,4)\) and \((4,3,10)\)

Find a vector that has the same direction as \(\langle- 2,4,2\rangle\) but has length \( 6 .\)

Find, correct to the nearest degree, the three angles of the triangle with vertices \(A(1,0,-1), B(3,-2,0),\) and \(C(1,3,3) .\)

A projectile is fired with an initial speed of 200 \(\mathrm{m} / \mathrm{s}\) and angle of elevation \(60^{\circ} .\) Find (a) the range of the projectile, (b) the maximum height reached, and (c) the speed at impact.

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

Find two unit vectors orthogonal to both \(\langle 3,2,1\rangle\) and \(\langle- 1,1,0\rangle\).