Chapter 11

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

Let \(\mathbf{a}\) and \(\mathbf{b}\) be nonparallel vectors, and let \(\mathbf{c}\) be any nonzero vector. Show that \((\mathbf{a} \times \mathbf{b}) \times \mathbf{c}\) is a vector in the plane of \(\mathbf{a}\) and \(\mathbf{b}\).

For what values of \(a, b\), and \(c\) are the three vectors \(\langle a, 0,1\rangle,\langle 0,2, b\rangle\), and \(\langle 1, c, 1\rangle\) mutually orthogonal.

A ship is sailing due south at 20 miles per hour. A man walks west (i.e., at right angles to the side of the ship) across the deck at 3 miles per hour. What are the magnitude and direction of his velocity relative to the surface of the water?

Find the general equation of a central ellipsoid that is symmetric with respect to the following: (a) origin (b) \(x\) -axis (c) \(x y\) -plane

Make the required change in the given equation. \(r^{2}+2 z^{2}=4\) to spherical coordinates

sketch the curve in the xy-plane. Then, for the given point, find the curvature and the radius of curvature. Finally, $$ y=\tan x,(\pi / 4,1) $$

Find the symmetric equations of the tangent line to the curve with equation $$ \mathbf{r}(t)=2 \cos t \mathbf{i}+6 \sin t \mathbf{j}+t \mathbf{k} $$ at \(t=\pi / 3\).

Find the volume of the parallelepiped with edges \(\langle 2,3,4\rangle,\langle 0,4,-1\rangle\), and \(\langle 5,1,3\rangle\) (see Example 4).

Find each of the given projections if \(\mathbf{u}=\mathbf{i}+2 \mathbf{j}, \mathbf{v}=2 \mathbf{i}-\mathbf{j}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}\). \(\operatorname{proj}_{\mathbf{v}} \mathbf{u}\)

Julie, flying in a wind blowing 40 miles per hour due south, discovers that she is heading due east when she points her airplane in the direction \(\mathrm{N} 60^{\circ} \mathrm{E}\). Find the airspeed (speed in still air) of the plane.

Find the general equation of a central hyperboloid of one sheet that is symmetric with respect to the following: (a) origin (b) \(y\) -axis (c) \(x y\) -plane

$$ \mathbf{r}(t)=\int_{1}^{t}\left[x^{2} \mathbf{i}+5(x-1)^{3} \mathbf{j}+(\sin \pi x) \mathbf{k}\right] d x ; t_{1}=2 $$

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

Find the parametric equations of the tangent line to the curve \(x=2 t^{2}, y=4 t, z=t^{3}\) at \(t=1\).

Find the volume of the parallelepiped with edges \(3 \mathbf{i}-4 \mathbf{j}+2 \mathbf{k},-\mathbf{i}+2 \mathbf{j}+\mathbf{k}\), and \(3 \mathbf{i}-2 \mathbf{j}+5 \mathbf{k}\).

Find each of the given projections if \(\mathbf{u}=\mathbf{i}+2 \mathbf{j}, \mathbf{v}=2 \mathbf{i}-\mathbf{j}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}\). \(\operatorname{proj}_{\mathbf{u}} \mathbf{v}\)

What heading and airspeed are required for an airplane to fly 837 miles per hour due north if a wind of 63 miles per hour is blowing in the direction \(\mathrm{S} 11.5^{\circ} \mathrm{E}\) ?

Find the general equation of a central hyperboloid of two sheets that is symmetric with respect to the following: (a) origin (b) \(z\) -axis (c) \(y z\) -plane

Find the velocity \(\mathbf{v}\), acceleration \(\mathbf{a}\), and speed \(s\) at the indicated time \(t=t_{1}\). $$ \mathbf{r}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+t \mathbf{k} ; t_{1}=\pi $$

sketch the curve in the xy-plane. Then, for the given point, find the curvature and the radius of curvature. Finally, $$ y=\sqrt[3]{x},(1,1) $$

Let \(K\) be the parallelepiped determined by \(\mathbf{u}=\langle 3,2,1\rangle, \mathbf{v}=\langle 1,1,2\rangle\), and \(\mathbf{w}=\langle 1,3,3\rangle .\) (a) Find the volume of \(K\). (b) Find the area of the face determined by \(\mathbf{u}\) and \(\mathbf{v}\). (c) Find the angle between \(\mathbf{u}\) and the plane containing the face determined by \(\mathbf{v}\) and \(\mathbf{w}\).

Find each of the given projections if \(\mathbf{u}=\mathbf{i}+2 \mathbf{j}, \mathbf{v}=2 \mathbf{i}-\mathbf{j}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}\). \(\operatorname{proj}_{\mathbf{u}} \mathbf{w}\)

Find the arc length of the given curve. \(x=t, y=t, z=2 t ; 0 \leq t \leq 2\)

Find the velocity \(\mathbf{v}\), acceleration \(\mathbf{a}\), and speed \(s\) at the indicated time \(t=t_{1}\). $$ \mathbf{r}(t)=\sin 2 t \mathbf{i}+\cos 3 t \mathbf{j}+\cos 4 t \mathbf{k} ; t_{1}=\frac{\pi}{2} $$

sketch the curve in the xy-plane. Then, for the given point, find the curvature and the radius of curvature. Finally, $$ y=\tanh x,\left(\ln 2, \frac{3}{5}\right) $$

Find the equation of the plane perpendicular to the curve $$ \mathbf{r}(t)=t \sin t \mathbf{i}+3 t \mathbf{j}+2 t \cos t \mathbf{k} $$ at \(t=\pi / 2\).

Find each of the given projections if \(\mathbf{u}=\mathbf{i}+2 \mathbf{j}, \mathbf{v}=2 \mathbf{i}-\mathbf{j}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}\). \(\operatorname{proj}_{\mathbf{u}}(\mathbf{w}-\mathbf{v})\)

Find the arc length of the given curve. \(x=t / 4, y=t / 3, z=t / 2 ; 1 \leq t \leq 3\)

If the curve \(z=x^{2}\) in the \(x z\) -plane is revolved about the \(z\) -axis, the resulting surface has equation \(z=x^{2}+y^{2}\), obtained as a result of replacing \(x\) by \(\sqrt{x^{2}+y^{2}}\). If \(y=2 x^{2}\) in the \(x y\) -plane is revolved about the \(y\) -axis, what is the equation of the resulting surface?

Find the velocity \(\mathbf{v}\), acceleration \(\mathbf{a}\), and speed \(s\) at the indicated time \(t=t_{1}\). $$ \mathbf{r}(t)=\tan t \mathbf{i}+3 e^{t} \mathbf{j}+\cos 4 t \mathbf{k} ; t_{1}=\frac{\pi}{4} $$

, find the curvature \(\kappa\), the unit tangent vector \(\mathbf{T}\), the unit normal vector \(\mathbf{N}\), and the binormal vector \(\mathbf{B}\) at \(t=t_{1} .\) $$ \mathbf{r}(t)=\frac{1}{2} t^{2} \mathbf{i}+t \mathbf{j}+\frac{1}{3} t^{3} \mathbf{k} ; t_{1}=2 $$

Consider the curve $$ \mathbf{r}(t)=2 t \mathbf{i}+\sqrt{7 t} \mathbf{j}+\sqrt{9-7 t-4 t^{2}} \mathbf{k}, 0 \leq t \leq \frac{1}{2} $$ (a) Show that the curve lies on a sphere centered at the origin. (b) Where does the tangent line at \(t=\frac{1}{4}\) intersect the \(x z\) -plane?

Which of the following do not make sense? (a) \(\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})\) (b) \(\mathbf{u}+(\mathbf{v} \times \mathbf{w})\) (c) \((\mathbf{a} \cdot \mathbf{b}) \times \mathbf{c}\) (d) \((\mathbf{a} \times \mathbf{b})+k\) (e) \((\mathbf{a} \cdot \mathbf{b})+k\) (f) \((\mathbf{a}+\mathbf{b}) \times(\mathbf{c}+\mathbf{d})\) (g) \((\mathbf{u} \times \mathbf{v}) \times \mathbf{w}\) (h) \((k \mathbf{u}) \times \mathbf{v}\)

Find each of the given projections if \(\mathbf{u}=\mathbf{i}+2 \mathbf{j}, \mathbf{v}=2 \mathbf{i}-\mathbf{j}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}\). \(\mathrm{pro} \mathrm{j}_{\mathrm{j}} \mathbf{u}\)

Prove, using vector methods, that the line segment joining the midpoints of two sides of a triangle is parallel to the third side.

Find the arc length of the given curve. \(x=t^{3 / 2}, y=3 t, z=4 t ; 1 \leq t \leq 4\)

Find the equation of the surface that results when the curve \(z=2 y\) in the \(y z\) -plane is revolved about the \(z\) -axis.

Make the required change in the given equation. \(r=2 \sin \theta\) to Cartesian coordinates

, find the curvature \(\kappa\), the unit tangent vector \(\mathbf{T}\), the unit normal vector \(\mathbf{N}\), and the binormal vector \(\mathbf{B}\) at \(t=t_{1} .\) $$ x=\sin 3 t, y=\cos 3 t, z=t, t_{1}=\pi / 9 $$

Consider the curve \(\mathbf{r}(t)=\sin t \cos t \mathbf{i}+\sin ^{2} t \mathbf{j}+\cos t \mathbf{k}\), \(0 \leq t \leq 2 \pi\) (a) Show that the curve lies on a sphere centered at the origin. (b) Where does the tangent line at \(t=\pi / 6\) intersect the \(x y\) -plane?

Find each of the given projections if \(\mathbf{u}=\mathbf{i}+2 \mathbf{j}, \mathbf{v}=2 \mathbf{i}-\mathbf{j}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}\). \(\operatorname{proj}_{\mathbf{i}} \mathbf{u}\)

Prove that the midpoints of the four sides of an arbitrarv quadrilateral are the vertices of a parallelogram.

Find the arc length of the given curve. \(x=t^{3 / 2}, y=t^{3 / 2}, z=t ; 2 \leq t \leq 4\)

Find the equation of the surface that results when the curve \(4 x^{2}+3 y^{2}=12\) in the \(x y\) -plane is revolved about the \(y\) -axis.

Find the velocity \(\mathbf{v}\), acceleration \(\mathbf{a}\), and speed \(s\) at the indicated time \(t=t_{1}\). $$ \mathbf{r}(t)=t \sin \pi t \mathbf{i}+t \cos \pi t \mathbf{j}+e^{-t} \mathbf{k} ; t_{1}=2 $$

Make the required change in the given equation. \(r^{2} \cos 2 \theta=z\) to Cartesian coordinates

, find the curvature \(\kappa\), the unit tangent vector \(\mathbf{T}\), the unit normal vector \(\mathbf{N}\), and the binormal vector \(\mathbf{B}\) at \(t=t_{1} .\) $$ x=7 \sin 3 t, y=7 \cos 3 t, z=14 t, t_{1}=\pi / 3 $$

Consider the curve \(\mathbf{r}(t)=2 t \mathbf{i}+t^{2} \mathbf{j}+\left(1-t^{2}\right) \mathbf{k}\) (a) Show that this curve lies on a plane and find the equation of this plane. (b) Where does the tangent line at \(t=2\) intersect the \(x y\) -plane?

The volume of a tetrahedron is known to be \(\frac{1}{3}(\) area of base )(height). From this, show that the volume of the tetrahedron with edges \(\mathbf{a}, \mathbf{b}\), and \(\mathbf{c}\) is \(\frac{1}{6}|\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})|\).