Problem 77
Find the distance from the sphere \(x^{2}+y^{2}+z^{2}+2 x+\) \(6 y-8 z=0\) to the plane \(3 x+4 y+z=15\).
Problem 78
Find the equation of the plane each of whose points is equidistant from \((-2,1,4)\) and \((6,1,-2)\).
Problem 79
Prove the Cauchy-Schwarz Inequality for two-dimensional vectors: $$ |\mathbf{u} \cdot \mathbf{v}| \leq\|\mathbf{u}\|\|\mathbf{v}\| $$
Problem 79
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.
Problem 81
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.
Problem 82
Show that the work done by a constant force \(\mathbf{F}\) on an object that moves completely around a closed polygonal path is \(0 .\)
Problem 83
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.
Problem 84
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.
Problem 85
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 \((6,1,2)\).
Problem 85
. 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 \(t=0\) )?
