Chapter 11

What condition leads to a graph that is symmetric with respect to the following? (a) \(x z\)-plane (b) \(y\)-axis (c) \(x\)-axis

Find the distance between the skew lines \(x=1+2 t\), \(y=-3+4 t, z=-1-t\) and \(x=4-2 t, y=1+3 t, z=2 t\) (see Problem 21).

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).

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 \(\mathrm{N} 60^{\circ}\) E. Find the airspeed (speed in still air) of the plane.

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

In Problems 23-28, find each of the given projections if \(\mathbf{u}=\mathbf{i}+2 \mathbf{j}, \mathbf{v}=2 \mathbf{i}-\mathbf{j}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}\). $$ \operatorname{proj}_{\mathbf{v}} \mathbf{u} $$

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

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 \(3 \mathbf{i}-4 \mathbf{j}+2 \mathbf{k},-\mathbf{i}+2 \mathbf{j}+\mathbf{k}\), and \(3 \mathbf{i}-2 \mathbf{j}+5 \mathbf{k}\)

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 \(\mathrm{S} 11.5^{\circ} \mathrm{E}\) ?

In Problems 17-30, make the required change in the given equation. \(\rho=2 \cos \phi\) to cylindrical coordinates

In Problems 23-28, find each of the given projections if \(\mathbf{u}=\mathbf{i}+2 \mathbf{j}, \mathbf{v}=2 \mathbf{i}-\mathbf{j}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}\). $$ \operatorname{proj}_{\mathbf{u}} \mathbf{v} $$

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

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

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

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

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 \(\mathbf{w}\).

In Problems 17-30, make the required change in the given equation. \(x+y=4\) to cylindrical coordinates

In Problems 23-28, find each of the given projections if \(\mathbf{u}=\mathbf{i}+2 \mathbf{j}, \mathbf{v}=2 \mathbf{i}-\mathbf{j}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}\). $$ \operatorname{proj}_{\mathbf{u}} \mathbf{w} $$

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

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

$$ \text { In Problems 25-32, find the arc length of the given curve. } $$ $$ x=t, y=t, z=2 t ; 0 \leq t \leq 2 $$

Find the equation of the plane perpendicular to the curve \(x=3 t, y=2 t^{2}, z=t^{5}\) at \(t=-1\)

In Problems 23-28, find each of the given projections if \(\mathbf{u}=\mathbf{i}+2 \mathbf{j}, \mathbf{v}=2 \mathbf{i}-\mathbf{j}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}\). $$ \operatorname{proj}_{\mathbf{u}}(\mathbf{w}-\mathbf{v}) $$

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

$$ \text { In Problems 25-32, find the arc length of the given curve. } $$ $$ x=t / 4, y=t / 3, z=t / 2 ; 1 \leq t \leq 3 $$

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) \((k \mathbf{u}) \times \mathbf{v}\)

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

In Problems 23-28, find each of the given projections if \(\mathbf{u}=\mathbf{i}+2 \mathbf{j}, \mathbf{v}=2 \mathbf{i}-\mathbf{j}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}\). $$ \operatorname{proj}_{\mathbf{j}} \mathbf{u} $$

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

$$ \text { In Problems 25-32, find the arc length of the given curve. } $$ $$ x=t^{3 / 2}, y=3 t, z=4 t ; 1 \leq t \leq 4 $$

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?

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

In Problems 17-30, make the required change in the given equation. \(r=2 \sin \theta\) to Cartesian coordinates

In Problems 23-28, find each of the given projections if \(\mathbf{u}=\mathbf{i}+2 \mathbf{j}, \mathbf{v}=2 \mathbf{i}-\mathbf{j}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}\). $$ \operatorname{proj}_{\mathbf{i}} \mathbf{u} $$

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

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.

$$ \text { In Problems 25-32, find the arc length of the given curve. } $$ $$ x=t^{3 / 2}, y=t^{3 / 2}, z=t ; 2 \leq t \leq 4 $$

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?

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 \(\frac{1}{6}|\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})|\).

In Problems 17-30, make the required change in the given equation. \(r^{2} \cos 2 \theta=z\) to Cartesian coordinates

In Problems 29-34, find each of the given projections if \(\mathbf{u}=3 \mathbf{i}+2 \mathbf{j}+\mathbf{k}, \mathbf{v}=2 \mathbf{i}-\mathbf{k}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}-3 \mathbf{k}\). $$ \operatorname{proj}_{\mathbf{v}} \mathbf{u} $$

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

$$ \mathbf{r}(t)=t \sin \pi t \mathbf{i}+t \cos \pi t \mathbf{j}+e^{-t} \mathbf{k}: t_{1}=2 $$

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.

$$ \text { In Problems 25-32, find the arc length of the given curve. } $$ $$ x=t^{2}, y=(4 / 3) t^{3 / 2}, z=t ; 0 \leq t \leq 8 $$

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?

Find the volume of the tetrahedron with vertices \((-1,2,3),(4,-1,2),(5,6,3)\), and \((1,1,-2)\) (see Problem 29\()\)

Let \(n\) points be equally spaced on a circle, and let \(\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{n}\) be the vectors from the center of the circle to these \(n\) points. Show that \(\mathbf{v}_{1}+\mathbf{v}_{2}+\cdots+\mathbf{v}_{n}=\mathbf{0}\).

In Problems 29-34, find each of the given projections if \(\mathbf{u}=3 \mathbf{i}+2 \mathbf{j}+\mathbf{k}, \mathbf{v}=2 \mathbf{i}-\mathbf{k}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}-3 \mathbf{k}\). $$ \operatorname{proj}_{\mathbf{u}} \mathbf{v} $$