Activity 9.6.3

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 $P_2$. (This is the same line as in Activity.)

1. Find parametric equations of the line $\mathcal{L}$.

2. Does the point $(1, 2, 1)$ lie on $\mathcal{L}$? If so, what value of $t$ results in this point?

3. Consider another line, $\mathcal{K}$, whose parametric equations are $$\begin{gathered} x(s) = 11 + 4s, \ \ y(s) = 1-3s, \ \ z(s) = 3 + 2s. \end{gathered}$$ What is the direction of the line $\mathcal{K}$?

4. Do the lines $\mathcal{L}$ and $\mathcal{K}$ intersect? If so, provide the point of intersection and the $t$ and $s$ values, respectively, that result in the point. If not, explain why. To find a point of intersection, you can set the coordinate equations of each line equal to each other try to solve for $t$ and $s$.