Chapter 11

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

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.

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

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.

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

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

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

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

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

Point to Line Let \(P\) be a point on a line with direction \(\mathbf{n}\) and \(Q\) a point off the line (Figure 7). Show that the distance \(d\) from \(Q\) to the line is given by $$ d=\frac{\|\overrightarrow{P Q} \times \mathbf{n}\|}{\|\mathbf{n}\|} $$ and use this result to find each distance in parts (a) and (b). (a) From \(Q(1,0,-4)\) to the line \(\frac{x-3}{2}=\frac{y+2}{-2}=\frac{z-1}{1}\) (b) From \(Q(2,-1,3)\) to the line \(x=1+2 t, y=-1+3 t\), \(z=-6 t\)

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.

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

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

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

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

$$ \text { In Problems 25-32, find the arc length of the given curve. } $$ $$ x=2 \cos t, y=2 \sin t, z=t / 20 ; 0 \leq t \leq 8 \pi $$

Line to Line 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 $$ d=\frac{|\overrightarrow{P Q} \cdot \mathbf{n}|}{\| \mathbf{n} \mid} $$ 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 \(x=3 t, y=1+t\), \(z=-5 t\)

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

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

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 33-36, 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

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

Find the great-circle distance from New York (longitude \(74^{\circ} \mathrm{W}\), latitude \(40.4^{\circ} \mathrm{N}\) ) to Greenwich (longitude \(0^{\circ}\), latitude \(51.3^{\circ} \mathrm{N}\) ).

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

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

In Problems 33-36, 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. $$ x=t, y=t^{2}, z=t^{3} ; 1 \leq t \leq 2 $$

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

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.

In Problems 33-36, 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. $$ x=2 \cos t, y=\sin t, z=t ; 0 \leq t \leq 6 \pi $$

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) \(\operatorname{proj}_{-\mathbf{u}}(-\mathbf{u})\)

Find the point of the curve at which the curvature is a maximum. \(y=\sin x ;-\pi \leq x \leq \pi\)

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.

In Problems 33-36, 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. $$ x=\sin t, y=\cos t, z=\sin t ; 0 \leq t \leq 2 \pi $$

Find the scalar projection of \(\mathbf{u}=-\mathbf{i}+5 \mathbf{j}+3 \mathbf{k}\) on \(\mathbf{v}=-\mathbf{i}+\mathbf{j}-\mathbf{k}\).

Find the point of the curve at which the curvature is a maximum. \(y=\cosh x\)

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 spiral \(\mathbf{r}=3 t \cos t \mathbf{i}+t \sin t \mathbf{j}+t \mathbf{k}\) lie?

Let vectors \(\mathbf{a}, \mathbf{b}\), and \(\mathbf{c}\) with common initial point determine a tetrahedron, and let \(\mathbf{m}, \mathbf{n}, \mathbf{p}\), and \(\mathbf{q}\) be vectors perpendicular to the four faces, pointing outward, and having length equal to the area of the corresponding face. Show that \(\mathbf{m}+\mathbf{n}+\mathbf{p}+\mathbf{q}=\mathbf{0}\)

Let \(\left(\rho_{1}, \theta_{1}, \phi_{1}\right)\) and \(\left(\rho_{2}, \theta_{2}, \phi_{2}\right)\) be the spherical coordinates of two points, and let \(d\) be the straight-line distance between them. Show that $$ \begin{aligned} d=\left\\{\left(\rho_{1}-\rho_{2}\right)^{2}+2 \rho_{1} \rho_{2}\left[1-\cos \left(\theta_{1}-\theta_{2}\right) \sin \phi_{1} \sin \phi_{2}\right.\right.\\\ &\left.\left.-\cos \phi_{1} \cos \phi_{2}\right]\right\\}^{1 / 2} \end{aligned} $$

Find the scalar projection of \(\mathbf{u}=5 \mathbf{i}+5 \mathbf{j}+2 \mathbf{k}\) on \(\mathbf{v}=-\sqrt{5} \mathbf{i}+\sqrt{5} \mathbf{j}+\mathbf{k}\)

Find the point of the curve at which the curvature is a maximum. \(y=\sinh x\)

Show that the curve determined by \(\mathbf{r}=t \mathbf{i}+t \mathbf{j}+t^{2} \mathbf{k}\) is a parabola, and find the coordinates of its focus.

Find the equations of the tangent spheres of equal radii whose centers are \((-3,1,2)\) and \((5,-3,6)\).

A vector \(\mathbf{u}=2 \mathbf{i}+3 \mathbf{j}+z \mathbf{k}\) emanating from the origin points into the first octant (i.e., that part of three-space where all components are positive). If \(\|\mathbf{u}\|=5\), find \(z\).

Find the point of the curve at which the curvature is a maximum. \(y=e^{x}\)

$$ \mathbf{f}(u)=\cos u \mathbf{i}+e^{3 u} \mathbf{j} \text { and } u(t)=3 t^{2}-4 $$

Find the equation of the sphere that is tangent to the three coordinate planes if its radius is 6 and its center is in the first octant.

As you may have guessed, there is a simple formula for expressing great-circle distance directly in terms of longitude and latitude. Let \(\left(\alpha_{1}, \beta_{1}\right)\) and \(\left(\alpha_{2}, \beta_{2}\right)\) be the longitude- latitude coordinates of two points on the surface of the earth, where we interpret \(\mathrm{N}\) and \(\mathrm{E}\) as positive and \(\mathrm{S}\) and \(\mathrm{W}\) as negative. Show that the great-circle distance between these points is \(3960 \gamma\) miles, where \(0 \leq \gamma \leq \pi\) and $$ \cos \gamma=\cos \left(\alpha_{1}-\alpha_{2}\right) \cos \beta_{1} \cos \beta_{2}+\sin \beta_{1} \sin \beta_{2} $$

If \(\alpha=46^{\circ}\) and \(\beta=108^{\circ}\) are direction angles for a vector \(\mathbf{u}\), find two possible values for the third angle.