Exercise 27

Recall that any line in space may be represented parametrically by a vector-valued function.

1. Find a vector-valued function $\vr$ that parameterizes the line through $(-2,1,4)$ in the direction of the vector $\vv = \langle 3, 2, -5 \rangle$.

2. Find a vector-valued function $\vr$ that parameterizes the line of intersection of the planes $x + 2y - z = 4$ and $3x + y - 2z = 1$.

3. Determine the point of intersection of the lines given by $$\begin{gathered} x = 2 + 3t, \ y = 1 - 2t, \ z = 4t, \end{gathered}$$ $$\begin{gathered} x = 3 + 1s, \ y = 3-2s, \ z = 2s. \end{gathered}$$ Then, find a vector-valued function $\vr$ that parameterizes the line that passes through the point of intersection you just found and is perpendicular to both of the given lines.