Chapter 11

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 arc length of the given curve. \(x=t^{2}, y=(4 / 3) t^{3 / 2}, z=t ; 0 \leq t \leq 8\)

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 \(x\) -axis.

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

Make the required change in the given equation. \(\rho \sin \phi=1\) 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 \(t=t_{1} .\) $$ \mathbf{r}(t)=\cos ^{3} t \mathbf{i}+\sin ^{3} t \mathbf{k} ; t_{1}=\pi / 2 $$

Let \(P\) be a point on a plane with normal vector \(\mathbf{n}\) and \(Q\) be a point off the plane. Show that the result of Example 10 of Section \(11.3\), the distance \(d\) between the point \(Q\) and the plane, can be expressed as $$ d=\frac{|\overrightarrow{P Q} \cdot \mathbf{n}|}{\|\mathbf{n}\|} $$ and use this result to find the distance from \((4,-2,3)\) to the plane \(4 x-4 y+2 z=2\).

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 \(\mathbf{v}_{1}+\mathbf{v}_{2}+\cdots+\mathbf{v}_{n}=\mathbf{0}\).

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 \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}-3 \mathbf{k}\). \(\operatorname{proj}_{\mathbf{u}} \mathbf{v}\)

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

Find the coordinates of the foci of the ellipse that is the intersection of \(z=x^{2} / 4+y^{2} / 9\) with the plane \(z=4\).

Show that if the speed of a moving particle is constant its acceleration vector is always perpendicular to its velocity vector.

The parabola \(z=2 x^{2}\) in the \(x z\) -plane is revolved about the \(z\) -axis. Write the equation of the resulting surface in cylindrical coordinates.

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

Prove Lagrange's Identity, $$ \|\mathbf{u} \times \mathbf{v}\|^{2}=\|\mathbf{u}\|^{2}\|\mathbf{v}\|^{2}-(\mathbf{u} \cdot \mathbf{v})^{2} $$ without using Theorem \(\mathrm{A}\).

Consider a horizontal triangular table with each vertex angle less than \(120^{\circ}\). At the vertices are frictionless pulleys over which pass strings knotted at \(P\), each with a weight \(W\) attached as shown in Figure \(20 .\) Show that at equilibrium the three angles at \(P\) are equal; that is, show that \(\alpha+\beta=\alpha+\gamma=\beta+\gamma=120^{\circ}\).

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 \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}-3 \mathbf{k}\). \(\operatorname{proj}_{\mathbf{u}} \mathbf{w}\)

Find the arc length of the given curve. \(x=2 \cos t, y=2 \sin t, z=3 t ;-\pi \leq t \leq \pi\)

Prove that \(\mid \mathbf{r}(t) \|\) is constant if and only if \(\mathbf{r}(t) \cdot \mathbf{r}^{\prime}(t)=0\).

The hyperbola \(2 x^{2}-z^{2}=2\) in the \(x z\) -plane is revolved about the \(z\) -axis. Write the equation of the resulting surface in cylindrical coordinates.

, find the curvature \(\kappa\), the unit tangent vector \(\mathbf{T}\), the unit normal vector \(\mathbf{N}\), and the binormal vector \(\mathbf{B}\) at \(t=t_{1} .\) $$ \mathbf{r}(t)=e^{7 t} \cos 2 t \mathbf{i}+e^{7 t} \sin 2 t \mathbf{j}+e^{7 t} \mathbf{k} ; t_{1}=\pi / 3 $$

Let \(P\) and \(Q\) be points on nonintersecting skew lines with directions \(\mathbf{n}_{1}\) and \(\mathbf{n}_{2}\), and let \(\mathbf{n}=\mathbf{n}_{1} \times \mathbf{n}_{2}\) (Figure 8). Show that the distance \(d\) between these lines is given by $$ d=\frac{|\overrightarrow{P Q} \cdot \mathbf{n}|}{\|\mathbf{n}\|} $$ and use this result to find the distance between each pair of lines in parts (a) and (b). (a) \(\frac{x-3}{1}=\frac{y+2}{1}=\frac{z-1}{2}\) and \(\frac{x+4}{3}=\frac{y+5}{4}=\frac{z}{5}\) (b) \(x=1+2 t, y=-2+3 t, z=-4 t\) and \(x=3 t, y=1+t\), \(z=-5 t\)

Prove the left distributive law, $$ \mathbf{u} \times(\mathbf{v}+\mathbf{w})=(\mathbf{u} \times \mathbf{v})+(\mathbf{u} \times \mathbf{w}) $$

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 \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}-3 \mathbf{k}\). \(\operatorname{proj}_{\mathbf{u}}(\mathbf{w}+\mathbf{v})\)

Find the arc length of the given curve. \(x=2 \cos t, y=2 \sin t, z=t / 20 ; 0 \leq t \leq 8 \pi\)

Find the area of the elliptical cross section cut from the surface \(x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1\) by the plane \(z=\) \(h,-c

In Problems \(33-38\), find the length of the curve with the given vector equation. $$ \mathbf{r}(t)=t \mathbf{i}+\sin t \mathbf{j}+\cos t \mathbf{k} ; 0 \leq t \leq 2 $$

, find the curvature \(\kappa\), the unit tangent vector \(\mathbf{T}\), the unit normal vector \(\mathbf{N}\), and the binormal vector \(\mathbf{B}\) at \(t=t_{1} .\) $$ \mathbf{r}(t)=e^{-2 t} \mathbf{i}+e^{2 t} \mathbf{j}+2 \sqrt{2} t \mathbf{k} ; t_{1}=0 $$

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 \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}-3 \mathbf{k}\). \(\operatorname{proj}_{\mathbf{k}} \mathbf{u}\)

Set up a definite integral for the arc length of the given curve. Use the Parabolic Rule with \(n=10\) or a CAS to approximate the integral. \(x=\sqrt{t}, y=t, z=t ; 1 \leq t \leq 6\)

Show that the volume of the solid bounded by the elliptic paraboloid \(x^{2} / a^{2}+y^{2} / b^{2}=h-z, h>0\), and the \(x y\) plane is \(\pi a b h^{2} / 2\), that is, the volume is one-half the area of the base times the height. Hint: Use the method of slabs of Section \(6.2\).

Find the length of the curve with the given vector equation. $$ \mathbf{r}(t)=t \cos t \mathbf{i}+t \sin t \mathbf{j}+\sqrt{2 t} \mathbf{k} ; 0 \leq t \leq 2 $$

, find the curvature \(\kappa\), the unit tangent vector \(\mathbf{T}\), the unit normal vector \(\mathbf{N}\), and the binormal vector \(\mathbf{B}\) at \(t=t_{1} .\) $$ x=\ln t, y=3 t, z=t^{2} ; t_{1}=2 $$

If both \(\mathbf{u} \times \mathbf{v}=\mathbf{0}\) and \(\mathbf{u} \cdot \mathbf{v}=0\), what can you conclude about u or \(\mathbf{v}\) ?

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 \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}-3 \mathbf{k}\). \(\operatorname{proj}_{\mathbf{i}} \mathbf{u}\)

Set up a definite integral for the arc length of the given curve. Use the Parabolic Rule with \(n=10\) or a CAS to approximate the integral. \(x=t, y=t^{2}, z=t^{3} ; 1 \leq t \leq 2\)

Show that the projection in the \(x z\) -plane of the curve that is the intersection of the surfaces \(y=4-x^{2}\) and \(y=x^{2}+z^{2}\) is an ellipse, and find its major and minor diameters.

Find the length of the curve with the given vector equation. $$ \mathbf{r}(t)=\sqrt{6} t^{2} \mathbf{i}+\frac{2}{3} t^{3} \mathbf{j}+6 t \mathbf{k} ; 3 \leq t \leq 6 $$

Find the great-circle distance from St. Paul (longitude \(93.1^{\circ} \mathrm{W}\), latitude \(\left.45^{\circ} \mathrm{N}\right)\) to Turin, Italy (longitude \(7.4^{\circ} \mathrm{E}\), latitude \(\left.45^{\circ} \mathrm{N}\right) .\)

A 100 -pound chandelier is held in place by four wires attached to the ceiling at the four corners of a square. Each wire makes an angle of \(45^{\circ}\) with the horizontal. Find the magnitude of the tension in each wire.

Find a simple expression for each of the following for an arbitrary vector \(\mathbf{u}\). (a) \(\operatorname{proj}_{\mathbf{u}} \mathbf{u}\) (b) \(\operatorname{proj}_{-\mathbf{u}} \mathbf{u}\)

Set up a definite integral for the arc length of the given curve. Use the Parabolic Rule with \(n=10\) or a CAS to approximate the integral. \(x=2 \cos t, y=\sin t, z=t ; 0 \leq t \leq 6 \pi\)

Sketch the triangle in the plane \(y=x\) that is above the plane \(z=y / 2\), below the plane \(z=2 y\), and inside the cylinder \(x^{2}+y^{2}=8\). Then find the area of this triangle.

Find the length of the curve with the given vector equation. $$ \mathbf{r}(t)=t^{2} \mathbf{i}-2 t^{3} \mathbf{j}+6 t^{3} \mathbf{k} ; 0 \leq t \leq 1 $$

Show that the triangle in the plane with vertices \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)\), and \(\left(x_{3}, y_{3}\right)\) has area equal to one-half the absolute value of the determinant $$ \left|\begin{array}{lll} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{array}\right| $$

Find a simple expression for each of the following for an arbitrary vector \(\mathbf{u}\). (a) \(\operatorname{proj}_{\mathbf{u}}(-\mathbf{u})\) (b) \(\operatorname{proj}_{-\mathbf{u}}(-\mathbf{u})\)

Set up a definite integral for the arc length of the given curve. Use the Parabolic Rule with \(n=10\) or a CAS to approximate the integral. \(x=\sin t, y=\cos t, z=\sin t ; 0 \leq t \leq 2 \pi\)

Show that the spiral \(\mathbf{r}=t \cos t \mathbf{i}+t \sin t \mathbf{j}+t \mathbf{k}\) lies on the circular cone \(x^{2}+y^{2}-z^{2}=0 .\) On what surface does the \(\operatorname{spiral} \mathbf{r}=3 t \cos t \mathbf{i}+t \sin t \mathbf{j}+t \mathbf{k}\) lie?

Find the length of the curve with the given vector equation. $$ \mathbf{r}(t)=t^{3} \mathbf{i}-2 t^{3} \mathbf{j}+6 t^{3} \mathbf{k} ; 0 \leq t \leq 1 $$