Chapter 11

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 equation of the sphere that has the line segment joining \((-2,3,6)\) and \((4,-1,5)\) as a diameter (see Example 3).

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.

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 $$ \begin{aligned} d=\left\\{\left(\rho_{1}-\rho_{2}\right)^{2}+2 \rho_{1} \rho_{2}\left[1-\cos \left(\theta_{1}-\theta_{2}\right)\right.\right.& \sin \phi_{1} \sin \phi_{2} \\\ -&\left.\left.\cos \phi_{1} \cos \phi_{2}\right]\right\\}^{1 / 2} \end{aligned} $$

, find the point of the curve at which the curvature is a maximum. $$ y=\sinh x $$

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 \(\mathbf{m}+\mathbf{n}+\mathbf{p}+\mathbf{q}=\mathbf{0}\).

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

\mathbf{F}(t)=\mathbf{f}(u(t)) .\( Find \)\mathbf{F}^{\prime}(t)\( in terms of \)t$ $$ \mathbf{f}(u)=\cos u \mathbf{i}+e^{3 \mu} \mathbf{j} \text { and } u(t)=3 t^{2}-4 $$

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 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.

\mathbf{F}(t)=\mathbf{f}(u(t)) .\( Find \)\mathbf{F}^{\prime}(t)\( in terms of \)t$ $$ \underline{\phantom{xxx}} \mathbf{f}(u)=u^{2} \mathbf{i}+\sin ^{2} u \mathbf{j} \text { and } u(t)=\tan t $$

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.

Find the equation of the sphere with center \((1,1,4)\) that is tangent to the plane \(x+y=12\).

Evaluate the integrals $$ \int_{0}^{1}\left(e^{\prime} \mathbf{i}+e^{-t \mathbf{j}}\right) d t $$

find the tangential and normal components \(\left(a_{T}\right.\) and \(\left.a_{N}\right)\) of the acceleration vector at \(t .\) Then evaluate at \(t=t_{1} .\) $$ \mathbf{r}(t)=3 t \mathbf{i}+3 t^{2} \mathbf{j} ; t_{1}=\frac{1}{3} $$

Find two perpendicular vectors \(\mathbf{u}\) and \(\mathbf{v}\) such that each is also perpendicular to \(\mathbf{w}=\langle-4,2,5\rangle\).

Describe the graph in three-space of each equation. (a) \(z=2\) (b) \(x=y\) (c) \(x y=0\) (d) \(x y z=0\) (e) \(x^{2}+y^{2}=4\) (f) \(z=\sqrt{9-x^{2}-y^{2}}\)

Evaluate the integrals $$ \int_{-1}^{1}\left[(1+t)^{3 / 2} \mathbf{i}+(1-t)^{3 / 2} \mathbf{j}\right] d t $$

find the tangential and normal components \(\left(a_{T}\right.\) and \(\left.a_{N}\right)\) of the acceleration vector at \(t .\) Then evaluate at \(t=t_{1} .\) $$ \mathbf{r}(t)=t^{2} \mathbf{i}+t \mathbf{j} ; t_{1}=1 $$

Find the vector emanating from the origin whose terminal point is the midpoint of the segment joining \((3,2,-1)\) and \((5,-7,2)\).

The sphere \((x-1)^{2}+(y+2)^{2}+(z+1)^{2}=10\) intersects the plane \(z=2\) in a circle. Find the circle's center and radius.

A point moves around the circle \(x^{2}+y^{2}=25\) at constant angular speed of 6 radians per second starting at \((5,0)\). Find expressions for \(\mathbf{r}(t), \mathbf{v}(t),\|\mathbf{v}(t)\|\), and $\mathbf{a}(t)

find the tangential and normal components \(\left(a_{T}\right.\) and \(\left.a_{N}\right)\) of the acceleration vector at \(t .\) Then evaluate at \(t=t_{1} .\) $$ \mathbf{r}(t)=(2 t+1) \mathbf{i}+\left(t^{2}-2\right) \mathbf{j} ; t_{1}=-1 $$

Which of the following do not make sense? (a) \(\mathbf{u} \cdot(\mathbf{v} \cdot \mathbf{w})\) (b) \((\mathbf{u} \cdot \mathbf{w})+\mathbf{w}\) \((\mathrm{c})\|\mathbf{u}\|(\mathbf{v} \cdot \mathbf{w})\) (d) \((\mathbf{u} \cdot \mathbf{v}) \mathbf{w}\)

An object's position \(P\) changes so that its distance from \((1,2,-3)\) is always twice its distance from \((1,2,3)\). Show that \(P\) is on a sphere and find its center and radius.

Consider the motion of a particle along a helix given by \(\mathbf{r}(t)=\sin t \mathbf{i}+\cos t \mathbf{j}+\left(t^{2}-3 t+2\right) \mathbf{k}\), where the \(\mathbf{k}\) component measures the height in meters above the ground and \(t \geq 0 .\) (a) Does the particle ever move downward? (b) Does the particle ever stop moving? (c) At what times does it reach a position 12 meters above the ground? (d) What is the velocity of the particle when it is 12 meters above the ground?

find the tangential and normal components \(\left(a_{T}\right.\) and \(\left.a_{N}\right)\) of the acceleration vector at \(t .\) Then evaluate at \(t=t_{1} .\) $$ \mathbf{r}(t)=a \cos t \mathbf{i}+a \sin t \mathbf{j} ; t_{1}=\pi / 6 $$

An object's position \(P\) changes so that its distance from \((1,2,-3)\) always equals its distance from \((2,3,2)\). Find the equation of the plane on which \(P\) lies.

In many places in the solar system, a moon orbits a planet, which in turn orbits the sun. In some cases the orbits are very close to circular. We will assume that these orbits are circular with the sun at the center of the planet's orbit and the planet at the center of the moon's orbit. We will further assume that all motion is in a single \(x y\) -plane. Suppose that in the time the planet orbits the sun once the moon orbits the planet ten times. (a) If the radius of the moon's orbit is \(R_{m}\) and the radius of the planet's orbit about the sun is \(R_{p}\), show that the motion of the moon with respect to the sun at the origin could be given by $$ x=R_{p} \cos t+R_{m} \cos 10 t, \quad y=R_{p} \sin t+R_{m} \sin 10 t $$ CAS (b) For \(R_{p}=1\) and \(R_{m}=0.1\), plot the path traced by the moon as the planet makes one revolution around the sun. (c) Find one set of values for \(R_{p}, R_{m}\) and \(t\) so that at time \(t\) the moon is motionless with respect to the sun.

find the tangential and normal components \(\left(a_{T}\right.\) and \(\left.a_{N}\right)\) of the acceleration vector at \(t .\) Then evaluate at \(t=t_{1} .\) $$ \mathbf{r}(t)=a \cosh t \mathbf{i}+a \sinh t \mathbf{j} ; t_{1}=\ln 3 $$

find the tangential and normal components \(\left(a_{T}\right.\) and \(\left.a_{N}\right)\) of the acceleration vector at \(t .\) Then evaluate at \(t=t_{1} .\) $$ x(t)=1+3 t, y(t)=2-6 t ; t_{1}=2 $$

Give a proof of the indicated property for two-dimensional vectors. Use \(\mathbf{u}=\left\langle u_{1}, u_{2}\right\rangle, \mathbf{v}=\left\langle v_{1}, v_{2}\right\rangle\), and \(\mathbf{w}=\left\langle w_{1}, w_{2}\right\rangle\). \(\mathbf{u} \cdot \mathbf{v}=\mathbf{v} \cdot \mathbf{u}\)

Describe in general terms the following "helical" type motions: (a) \(\mathbf{r}(t)=\sin t \mathbf{i}+\cos t \mathbf{j}+t \mathbf{k}\) (b) \(\mathbf{r}(t)=\sin t^{3} \mathbf{i}+\cos t^{3} \mathbf{j}+t^{3} \mathbf{k}\) (c) \(\mathbf{r}(t)=\sin \left(t^{3}+\pi\right) \mathbf{i}+t^{3} \mathbf{j}+\cos \left(t^{3}+\pi\right) \mathbf{k}\) (d) \(\mathbf{r}(t)=t \sin t \mathbf{i}+t \cos t \mathbf{j}+t \mathbf{k}\) (e) \(\mathbf{r}(t)=t^{-2} \sin t \mathbf{i}+t^{-2} \cos t \mathbf{j}+t \mathbf{k}, t>0\) (f) \(\mathbf{r}(t)=t^{2} \sin (\ln t) \mathbf{i}+\ln t \mathbf{j}+t^{2} \cos (\ln t) \mathbf{k}, t>1\)

find the tangential and normal components \(\left(a_{T}\right.\) and \(\left.a_{N}\right)\) of the acceleration vector at \(t .\) Then evaluate at \(t=t_{1} .\) $$ \mathbf{r}(t)=(t+1) \mathbf{i}+3 t \mathbf{j}+t^{2} \mathbf{k} ; t_{1}=1 $$

The curve defined by \(x=a \cos t, y=a \sin t, z=c t\) is a helix. Hold \(a\) fixed and use a CAS to obtain a parmetric plot of the helix for various values of \(c .\) What effect does \(c\) have on the curve?

EXPL 48. In this exercise you will derive Kepler's First Law, that planets travel in elliptical orbits. We begin with the notation. Place the coordinate system so that the sun is at the origin and the planet's closest approach to the sun (the perihelion) is on the positive \(x\) -axis and occurs at time \(t=0\). Let \(\mathbf{r}(t)\) denote the position vector and let \(r(t)=\|\mathbf{r}(t)\|\) denote the distance from the sun at time \(t\). Also, let \(\theta(t)\) denote the angle that the vector \(\mathbf{r}(t)\) makes with the positive \(x\) -axis at time \(t\). Thus, \((r(t), \theta(t))\) is the polar coordinate representation of the planet's position. Let \(\mathbf{u}_{1}=\mathbf{r} / r=(\cos \theta) \mathbf{i}+(\sin \theta) \mathbf{j} \quad\) and \(\quad \mathbf{u}_{2}=(-\sin \theta) \mathbf{i}+(\cos \theta) \mathbf{j}\) Vectors \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2}\) are orthogonal unit vectors pointing in the directions of increasing \(r\) and increasing \(\theta\), respectively. Figure 12 summarizes this notation. We will often omit the argument \(t\), but keep in mind that \(\mathbf{r}, \theta, \mathbf{u}_{1}\), and \(\mathbf{u}_{2}\) are all functions of \(t .\) A prime in. dicates differentiation with respect to time \(t\). (a) Show that \(\mathbf{u}_{1}^{\prime}=\theta^{\prime} \mathbf{u}_{2}\) and \(\mathbf{u}_{2}^{\prime}=-\theta^{\prime} \mathbf{u}_{1}\). (b) Show that the velocity and acceleration vectors satisfy $$ \begin{array}{l} \mathbf{v}=r^{\prime} \mathbf{u}_{1}+r \theta^{\prime} \mathbf{u}_{2} \\ \mathbf{a}=\left(r^{\prime \prime}-r\left(\theta^{\prime}\right)^{2}\right) \mathbf{u}_{1}+\left(2 r^{\prime} \theta^{\prime}+r \theta^{\prime \prime}\right) \mathbf{u}_{2} \end{array} $$ (c) Use the fact that the only force acting on the planet is the gravity of the sun to express a as a multiple of \(\mathbf{u}_{1}\), then explain how we can conclude that $$ \begin{aligned} r^{\prime \prime}-r\left(\theta^{\prime}\right)^{2} &=\frac{-G M}{r^{2}} \\ 2 r^{\prime} \theta^{\prime}+r \theta^{\prime \prime} &=0 \end{aligned} $$ (d) Consider \(\mathbf{r} \times \mathbf{r}^{\prime}\), which we showed in Example 8 was a constant vector, say D. Use the result from (b) to show that \(\mathbf{D}=r^{2} \theta^{\prime} \mathbf{k} .\) (e) Substitute \(t=0\) to get \(\mathbf{D}=r_{0} v_{0} \mathbf{k}\), where \(r_{0}=r(0)\) and \(v_{0}=\|\mathbf{v}(0)\|\). Then argue that \(r^{2} \theta^{\prime}=r_{0} v_{0}\) for all \(t\). (f) Make the substitution \(q=r^{\prime}\) and use the result from (e) to obtain the first-order (nonlinear) differential equation in \(q\) : $$ q \frac{d q}{d r}=\frac{r_{0}^{2} v_{0}^{2}}{r^{3}}-\frac{G M}{r^{2}} $$ (g) Integrate with respect to \(r\) on both sides of the above equation and use an initial condition to obtain $$ q^{2}=2 G M\left(\frac{1}{r}-\frac{1}{r_{0}}\right)+v_{0}^{2}\left(1-\frac{r_{0}^{2}}{r^{2}}\right) $$ (h) Substitute \(p=1 / r\) into the above equation to obtain $$ \frac{r_{0}^{2} v_{0}^{2}}{\left(\theta^{\prime}\right)^{2}}\left(\frac{d p}{d t}\right)^{2}=2 G M\left(p-p_{0}\right)+v_{0}^{2}\left(1-\frac{p^{2}}{p_{0}^{2}}\right) $$

find the tangential and normal components \(\left(a_{T}\right.\) and \(\left.a_{N}\right)\) of the acceleration vector at \(t .\) Then evaluate at \(t=t_{1} .\) $$ x=t, y=t^{2}, z=t^{3} ; t_{1}=2 $$

Give a proof of the indicated property for two-dimensional vectors. Use \(\mathbf{u}=\left\langle u_{1}, u_{2}\right\rangle, \mathbf{v}=\left\langle v_{1}, v_{2}\right\rangle\), and \(\mathbf{w}=\left\langle w_{1}, w_{2}\right\rangle\). \(\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})=\mathbf{u} \cdot \mathbf{v}+\mathbf{u} \cdot \mathbf{w}\)

find the tangential and normal components \(\left(a_{T}\right.\) and \(\left.a_{N}\right)\) of the acceleration vector at \(t .\) Then evaluate at \(t=t_{1} .\) $$ x=e^{-t}, y=2 t, z=e^{t} ; t_{1}=0 $$

Give a proof of the indicated property for two-dimensional vectors. Use \(\mathbf{u}=\left\langle u_{1}, u_{2}\right\rangle, \mathbf{v}=\left\langle v_{1}, v_{2}\right\rangle\), and \(\mathbf{w}=\left\langle w_{1}, w_{2}\right\rangle\). \(\mathbf{0} \cdot \mathbf{u}=0\)

find the tangential and normal components \(\left(a_{T}\right.\) and \(\left.a_{N}\right)\) of the acceleration vector at \(t .\) Then evaluate at \(t=t_{1} .\) $$ \mathbf{r}(t)=(t-2)^{2} \mathbf{i}-t^{2} \mathbf{j}+t \mathbf{k} ; t_{1}=2 $$

Give a proof of the indicated property for two-dimensional vectors. Use \(\mathbf{u}=\left\langle u_{1}, u_{2}\right\rangle, \mathbf{v}=\left\langle v_{1}, v_{2}\right\rangle\), and \(\mathbf{w}=\left\langle w_{1}, w_{2}\right\rangle\). \(\mathbf{u} \cdot \mathbf{u}=\|\mathbf{u}\|^{2}\)

find the tangential and normal components \(\left(a_{T}\right.\) and \(\left.a_{N}\right)\) of the acceleration vector at \(t .\) Then evaluate at \(t=t_{1} .\) $$ \mathbf{r}(t)=\left(t-\frac{1}{3} t^{3}\right) \mathbf{i}-\left(t+\frac{1}{3} t^{3}\right) \mathbf{j}+t \mathbf{k} ; t_{1}=3 $$

Given the two nonparallel vectors \(\mathbf{a}=3 \mathbf{i}-2 \mathbf{j}\) and \(\mathbf{b}=-3 \mathbf{i}+4 \mathbf{j}\) and another vector \(\mathbf{r}=7 \mathbf{i}-8 \mathbf{j}\), find scalars \(k\) and \(m\) such that \(\mathbf{r}=k \mathbf{a}+m \mathbf{b}\).

find the tangential and normal components \(\left(a_{T}\right.\) and \(\left.a_{N}\right)\) of the acceleration vector at \(t .\) Then evaluate at \(t=t_{1} .\) $$ \mathbf{r}(t)=t \mathbf{i}+\frac{1}{3} t^{3} \mathbf{j}+t^{-1} \mathbf{k}, t>0 ; t_{1}=1 $$

Given the two nonparallel vectors \(\mathbf{a}=-4 \mathbf{i}+3 \mathbf{j}\) and \(\mathbf{b}=2 \mathbf{i}-\mathbf{j}\) and another vector \(\mathbf{r}=6 \mathbf{i}-7 \mathbf{j}\), find scalars \(k\) and \(m\) such that \(\mathbf{r}=k \mathbf{a}+m \mathbf{b}\).

. Sketch the path for a particle if its position vector is \(\mathbf{r}=\sin t \mathbf{i}+\sin 2 t \mathbf{j}, 0 \leq t \leq 2 \pi\) (you should get a figure eight). Where is the acceleration zero? Where does the acceleration vector point to the origin?

Show that the vector \(\mathbf{n}=a \mathbf{i}+b \mathbf{j}\) is perpendicular to the line with equation \(a x+b y=c .\) Hint: Let \(P_{1}\left(x_{1}, y_{1}\right)\) and \(P_{2}\left(x_{2}, y_{2}\right)\) be two points on the line and show that \(\mathbf{n} \cdot \overrightarrow{P_{1} P_{2}}=0\).