Problem 223
Write the parametric equations of the line through…
Write the parametric equations of the line through $M_1(1,0,2)$ parallel to $\vec p=\{-1,3,2\}$.
Explanation of Method
The parametric equations of the line through M₁(1, 0, 2) with a given direction.
Step 1: Solution
\[x=1-t,\qquad y=3t,\qquad z=2+2t\qquad(t\in\mathbb R)\]
Canonical form: $\frac{x-1}{-1}=\frac y3=\frac{z-2}{2}$.
$t=0$ gives $M_1$. For $t=1$ we get the point $(0,3,4)$, where the line crosses the plane $yOz$.
Result
\[\boxed{\,x=1-t,\qquad y=3t,\qquad z=2+2t\qquad(t\in\mathbb R)\,}\]