Chapter 11

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.

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.

In Problems 65-68, find the equation of the plane having the given normal vector \(\mathbf{n}\) and passing through the given point \(P\). $$ \mathbf{n}=2 \mathbf{i}-4 \mathbf{j}+3 \mathbf{k} ; P(1,2,-3) $$

In Problems 65-68, find the equation of the plane having the given normal vector \(\mathbf{n}\) and passing through the given point \(P\). $$ \mathbf{n}=3 \mathbf{i}-2 \mathbf{j}-1 \mathbf{k} ; P(-2,-3,4) $$

Show that the curve $$ y=\left\\{\begin{array}{rll} 0 & \text { if } & x \leq 0 \\ x^{3} & \text { if } & x>0 \end{array}\right. $$ has continuous first derivatives and curvature at all points.

In Problems 65-68, 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 \(x \geq 1\).

Derive the polar coordinate curvature formula $$ \kappa=\frac{\left|r^{2}+2\left(r^{\prime}\right)^{2}-r r^{\prime \prime}\right|}{\left(r^{2}+\left(r^{\prime}\right)^{2}\right)^{3 / 2}} $$ where the derivatives are with respect to \(\theta\).

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 equation of the plane passing through the origin and parallel to (a) the \(x y\)-plane (b) the plane \(x+y+z=1\)

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 \(r=e^{69}\) is proportional to \(1 / r\).

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

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

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

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

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.

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 unit binormal vector \(\mathbf{B}=\mathbf{T} \times \mathbf{N}\) has the property that \(\frac{d \mathbf{B}}{d s}\) is perpendicular to \(\mathbf{T}\).

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.

Refine the method of Example 10 by showing that the distance \(L\) between the parallel planes \(A x+B y+C z=D\) and \(A x+B y+C z=E\) is $$ L=\frac{|D-E|}{\sqrt{A^{2}+B^{2}+C^{2}}} $$

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.

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 \((6,1,2)\).

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 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.