Chapter 11

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

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?

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

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

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

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

Find the angles formed by the diagonals of a cube.

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 \(\mathbf{a}\) 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 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).

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.

Find the work done by a force \(\mathbf{F}=-5 \mathbf{i}+8 \mathbf{j}\) newtons in moving an object 12 meters north.

Find the work done by a force \(\mathbf{F}=-4 \mathbf{k}\) newtons in moving an object from \((0,0,8)\) to \((4,4,0)\), where distance is in meters.

Show that for a plane curve \(\mathbf{N}\) points to the concave side of the curve. Hint: One method is to show that $$ \mathbf{N}=(-\sin \phi \mathbf{i}+\cos \phi \mathbf{j}) \frac{d \phi / d s}{|d \phi / d s|} $$ Then consider the cases \(d \phi / d s>0\) (curve bends to the left) and \(d \phi / d s<0\) (curve bends to the right).

Find the work done by a force \(\mathbf{F}=3 \mathbf{i}-6 \mathbf{j}+7 \mathbf{k}\) pounds in moving an object from \((2,1,3)\) to \((9,4,6)\), where distance is in feet.

Find the equation of the plane having the given normal vector \(\mathbf{n}\) and passing through the given point \(P .\) $$ \mathbf{n}=2 \mathbf{i}-4 \mathbf{j}+3 \mathbf{k} ; P(1,2,-3) $$

Find the equation of the plane having the given normal vector \(\mathbf{n}\) and passing through the given point \(P .\) $$ \mathbf{n}=3 \mathbf{i}-2 \mathbf{j}-1 \mathbf{k} ; P(-2,-3,4) $$

. Find a curve given by a polynominal \(P_{5}(x)\) that provides a smooth transition between two horizontal lines. That is, assume a function of the form \(P_{5}(x)=a_{0}+a_{1} x+a_{2} x^{2}+\) \(a_{3} x^{3}+a_{4} x^{4}+a_{5} x^{5}\), which provides a smooth transition between \(y=0\) for \(x \leq 0\) and \(y=1\) for \(x \geq 1\) in such a way that the function, its derivative, and curvature are all continuous for all values of \(x\). $$ y=\left\\{\begin{array}{ll} 0 & \text { if } \quad x \leq 0 \\ P_{5}(x) & \text { if } \quad 0

Find the equation of the plane having the given normal vector \(\mathbf{n}\) and passing through the given point \(P .\) $$ \mathbf{n}=\langle 1,4,4\rangle ; P(1,2,1) $$

Find a curve given by a polynomial \(P_{5}(x)\) that provides a smooth transition between \(y=0\) for \(x \leq 0\) and \(y=x\) for \(x \geq 1\)

Find the equation of the plane having the given normal vector \(\mathbf{n}\) and passing through the given point \(P .\) $$ \mathbf{n}=\langle 0,0,1\rangle ; P(1,2,-3) $$

Find the equation of the plane through \((-1,2,-3)\) and parallel to the plane \(2 x+4 y-z=6\).

Find the equation of the plane passing through \((-4,-1,2)\) and parallel to (a) the \(x y\) -plane (b) the plane \(2 x-3 y-4 z=0\)

Find the distance from \((1,-1,2)\) to the plane \(x+3 y+z=7\).

Find the distance from \((2,6,3)\) to the plane \(-3 x+2 y+z=9\).

Find the distance between the parallel planes \(-3 x+2 y+z=9\) and \(6 x-4 y-2 z=19\).

. Show that the curvature of the polar curve \(r=e^{6 \theta}\) is proportional to \(1 / r\).

Find the distance between the parallel planes \(5 x-3 y-2 z=5\) and \(-5 x+3 y+2 z=7\).

Show that the curvature of the polar curve \(r^{2}=\cos 2 \theta\) is directly proportional to \(r\) for \(r>0\).

Find the distance from the sphere \(x^{2}+y^{2}+z^{2}+2 x+\) \(6 y-8 z=0\) to the plane \(3 x+4 y+z=15\).

Find the equation of the plane each of whose points is equidistant from \((-2,1,4)\) and \((6,1,-2)\).

Draw the graph of \(x=4 \cos t, y=3 \sin (t+0.5)\), \(0 \leq t \leq 2 \pi .\) Estimate its maximum and minimum curvature by looking at the graph (curvature is the reciprocal of the radius of curvature). Then use a graphing calculator or a CAS to approximate these two numbers to four decimal places.

Prove the Cauchy-Schwarz Inequality for two-dimensional vectors: $$ |\mathbf{u} \cdot \mathbf{v}| \leq\|\mathbf{u}\|\|\mathbf{v}\| $$

A weight of 30 pounds is suspended by three wires with resulting tensions \(3 \mathbf{i}+4 \mathbf{j}+15 \mathbf{k},-8 \mathbf{i}-2 \mathbf{j}+10 \mathbf{k}\), and \(a \mathbf{i}+b \mathbf{j}+c \mathbf{k} .\) Determine \(a, b\), and \(c\) so that the net force is straight up.

Show that the work done by a constant force \(\mathbf{F}\) on an object that moves completely around a closed polygonal path is \(0 .\)

Let \(\mathbf{a}=\left\langle a_{1}, a_{2}, a_{3}\right\rangle\) and \(\mathbf{b}=\left\langle b_{1}, b_{2}, b_{3}\right\rangle\) be fixed vectors. Show that \((\mathbf{x}-\mathbf{a}) \cdot(\mathbf{x}-\mathbf{b})=0\) is the equation of a sphere, and find its center and radius.

Show that for a straight line \(\mathbf{r}(t)=\mathbf{r}_{0}+a_{0} t \mathbf{i}+\) \(b_{0} t \mathbf{j}+c_{0} t \mathbf{k}\) both \(\kappa\) and \(\tau\) are zero.

. A fly is crawling along a wire helix so that its position vector is \(\mathbf{r}(t)=6 \cos \pi t \mathbf{i}+6 \sin \pi t \mathbf{j}+2 t \mathbf{k}, t \geq 0 .\) At what point will the fly hit the sphere \(x^{2}+y^{2}+z^{2}=100\), and how far did it travel in getting there (assuming that it started when \(t=0\) )?

The medians of a triangle meet at a point \(P\) (the centroid by Problem 30 of Section 6.6) that is two-thirds of the way from a vertex to the midpoint of the opposite edge. Show that \(P\) is the head of the position vector \((\mathbf{a}+\mathbf{b}+\mathbf{c}) / 3\), where \(\mathbf{a}, \mathbf{b}\), and \(\mathbf{c}\) are the position vectors of the vertices, and use this to find \(P\) if the vertices are \((2,6,5),(4,-1,2)\), and \((6,1,2)\).

The DNA molecule in humans is a double helix, each with about \(2.9 \times 10^{8}\) complete turns. Each helix has radius about 10 angstroms and rises about 34 angstroms on each complete turn (an angstrom is \(10^{-8}\) centimeter). What is the total length of such a helix?

Suppose that the three coordinate planes bounding the first octant are mirrors. A light ray with direction \(a \mathbf{i}+b \mathbf{j}+c \mathbf{k}\) is reflected successively from the \(x y\) -plane, the \(x z\) -plane, and the \(y z\) -plane. Determine the direction of the ray after each reflection, and state a nice conclusion concerning the final reflected ray.