91Ó°ÊÓ

Geometry Using vectors, prove that the diagonals of a parallelogram bisect each other.

Sketch the solid that has the given description in cylindrical coordinates. \(0 \leq \theta \leq \pi / 2,0 \leq r \leq 2,0 \leq z \leq 4\)

Sketch the solid that has the given description in spherical coordinates. \(0 \leq \theta \leq 2 \pi, \pi / 4 \leq \phi \leq \pi / 2,0 \leq \rho \leq 1\)

Distance Two insects are crawling along different lines in three-space. At time \(t\) (in minutes), the first insect is at the point \((x, y, z)\) on the line \(x=6+t, \quad y=8-t, \quad z=3+t\) Also, at time \(t\), the second insect is at the point \((x, y, z)\) on the line \(x=1+t, \quad y=2+t, \quad z=2 t\) Assume distances are given in inches. (a) Find the distance between the two insects at time \(t=0\). (b) Use a graphing utility to graph the distance between the insects from \(t=0\) to \(t=10\). (c) Using the graph from part (b), what can you conclude about the distance between the insects? (d) How close do the insects get?

The spherical coordinates of a point \((x, y, z)\) are unique.Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.