Exercise 24

The vector and parametric forms of a line allow us to easily describe line segments in space.

Let $P_1 = (1,2,-1)$ and $P_2 = (-2,1,-2)$, and let $\mathcal{L}$ be the line in $\R^3$ through $P_1$ and $P_2$ as in Activity.

1. What value of the parameter $t$ makes $(x(t), y(t), z(t)) = P_1$? What value of $t$ makes $(x(t), y(t), z(t)) = P_2$?

2. What $t$ values describe the line segment between the points $P_1$ and $P_2$?

3. What about the line segment (along the same line) from $(7,4,1)$ to $(-8,-1,-4)$?

4. Now, consider a segment that lies on a different line: parameterize the segment that connects point $R=(4,-2,7)$ to $Q=(-11,4,27)$ in such a way that $t = 0$ corresponds to point $Q$, while $t = 2$ corresponds to $R$.