Exercise 25

This exercise explores key relationships between a pair of lines. Consider the following two lines: one with parametric equations $x(s) = 4-2s$, $y(s) = -2 + s$, $z(s) = 1 + 3s$, and the other being the line through $(-4, 2, 17)$ in the direction $\vv = \langle -2, 1, 5 \rangle$.

1. Find a direction vector for the first line, which is given in parametric form.

2. Find parametric equations for the second line, written in terms of the parameter $t$.

3. Show that the two lines intersect at a single point by finding the values of $s$ and $t$ that result in the same point. Then find the point of intersection.

4. Find the acute angle formed where the two lines intersect, noting that this angle will be given by the acute angle between their respective direction vectors.