Chapter 11

Find the point of the curve at which the curvature is a maximum. \(y=\ln \cos x\) for \(-\pi / 2

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

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

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

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

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) \(z=\sqrt{9-x^{2}-y^{2}}\)

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

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

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

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.

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

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

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 a \((t)\) (see Example 3 ).

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.

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

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?

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.

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

The solid spheres \((x-1)^{2}+(y-2)^{2}+(z-1)^{2} \leq 4\) and \((x-2)^{2}+(y-4)^{2}+(z-3)^{2} \leq 4\) intersect in a solid. Find its volume.

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

In Problems 45-50, 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 \(\mathbf{w}=\left\langle w_{1}, w_{2}\right\rangle\). $$ c(\mathbf{u} \cdot \mathbf{v})=(c \mathbf{u}) \cdot \mathbf{v} $$

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

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) \(\mathbf{r}(t)=t^{2} \sin (\ln t) \mathbf{i}+\ln t \mathbf{j}+t^{2} \cos (\ln t) \mathbf{k}, t>1\)

47\. 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?

In Problems 45-50, 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 \(a_{N}\) ) of the acceleration vector at \(t\). Then evaluate at \(t=t_{1}\). \(x=t, y=t^{2}, z=t^{3} ; t_{1}=2\)

In Problems 45-50, 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 \(a_{N}\) ) of the acceleration vector at \(t\). Then evaluate at \(t=t_{1}\). \(x=e^{-t}, y=2 t, z=e^{t} ; t_{1}=0\)

In Problems 45-50, 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 \(a_{N}\) ) of the acceleration vector at \(t\). Then evaluate at \(t=t_{1}\). \(\mathbf{r}(t)=(t-2)^{2} \mathbf{i}-t^{2} \mathbf{j}+t \mathbf{k} ; t_{1}=2\)

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 \(a_{N}\) ) 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 \(\mathbf{a}=-4 \mathbf{i}+3 \mathbf{j}\) 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 \(\mathbf{r}=k \mathbf{a}+m \mathbf{b}\).

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

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 \(\mathbf{n} \cdot \overrightarrow{P_{1} P_{2}}=0\).

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?

Prove that \(\|\mathbf{u}+\mathbf{v}\|^{2}+\|\mathbf{u}-\mathbf{v}\|^{2}=2\|\mathbf{u}\|^{2}+2\|\mathbf{v}\|^{2}\).

The position vector of a particle at time \(t \geq 0\) is $$ \mathbf{r}(t)=(\cos t+t \sin t) \mathbf{i}+(\sin t-t \cos t) \mathbf{j} $$ (a) Show that the speed \(d s / d t=t\). (b) Show that \(a_{T}=1\) and \(a_{N}=t\).

Prove that \(\mathbf{u} \cdot \mathbf{v}=\frac{1}{4}\|\mathbf{u}+\mathbf{v}\|^{2}-\frac{1}{4}\|\mathbf{u}-\mathbf{v}\|^{2}\).

If, for a particle, \(a_{T}=0\) for all \(t\), what can you conclude about its speed? If \(a_{N}=0\) for all \(t\), what can you conclude about its curvature?

Find the angle between a main diagonal of a cube and one of its faces.

Find the smallest angle between the main diagonals of a rectangular box 4 feet by 6 feet by 10 feet.

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 \(t \geq 0\). If the particle leaves the helix and moves along the line tangent to the helix when it is 12 meters above the ground, give the direction vector for the line.

Find the angles formed by the diagonals of a cube.

An object moves along the curve \(y=\sin 2 x\). Without doing any calculating, decide where \(a_{N}=0\).

Find the work done by the force \(\mathbf{F}=3 \mathbf{i}+10 \mathbf{j}\) newtons in moving an object 10 meters north (i.e., in the \(\mathbf{j}\) direction).

A dog is running counterclockwise around the circle \(x^{2}+y^{2}=400\) (distances in feet). At the point \((-12,16)\), it is running at 10 feet per second and is speeding up at 5 feet per second per second. Express its acceleration a at the point first in terms of \(\mathbf{T}\) and \(\mathbf{N}\), and then in terms of \(\mathbf{i}\) and \(\mathbf{j}\).

Find the work done by a force of 100 newtons acting in the direction \(S 70^{\circ} \mathrm{E}\) in moving an object 30 meters east.

Find the work done by the force \(\mathbf{F}=6 \mathbf{i}+8 \mathbf{j}\) pounds in moving an object from \((1,0)\) to \((6,8)\), where distance is in feet.

A car traveling at constant speed \(v\) rounds a level curve, which we take to be a circle of radius \(R\). If the car is to avoid sliding outward, the horizontal frictional force \(F\) exerted by the road on the tires must at least balance the centrifugal force pulling outward. The force \(F\) satisfies \(F=\mu m g\), where \(\mu\) is the coefficient of friction, \(m\) is the mass of the car, and \(g\) is the acceleration of gravity. Thus, \(\mu m g \geq m v^{2} / R\). Show that \(v_{R}\), the speed beyond which skidding will occur, satisfies $$ v_{R}=\sqrt{\mu g R} $$ and use this to determine \(v_{R}\) for a curve with \(R=400\) feet and \(\mu=0.4\). Use \(g=32\) feet per second per second.