Activity 9.6.2

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 $P_2$. Note that Figure shows a similar example of a line in 3D defined by two points.

1. Find a direction vector for the line $\mathcal{L}$.

2. Find a vector equation of $\mathcal{L}$ in the form $\vr(t) = \vr_0 + t\vv$.

3. Consider the vector equation $\vs(t) = \langle -5, 0, -3 \rangle + t \langle 6, 2, 2 \rangle.$ What is the direction of the line given by $\vs(t)$? Is this new line parallel to line $\mathcal{L}$?

4. Do $\vr(t)$ and $\vs(t)$ represent the same line, $\mathcal{L}$? Write a couple of sentences to justify why you think $\vr(t)$ and $\vs(t)$ do or do not describe the same set of points.