Answer
$r(t) =\lt -1-2t,2+3t,-2+3t \gt$ and $ \\0 \leq t \leq 1$
$x=-1-2t\\ y= 2+3t; \\z=-2+3t ; \\0 \leq t \leq 1$
Work Step by Step
The vector line equation of a line segment joining the points with position vectors $r_0$ and $r_1$ for the given two points is as follows:
$r(t)=(1-t) r_0+tr_1=(1-t) \lt -1, 2,-2 \gt +t \lt -3, 5,1 \gt$
or, $=\lt -1+t,2-2t, -2+2t \gt + \lt -3t,5t,t \gt$
or, $=\lt -1-2t,2+3t,-2+3t \gt$
Thus, we have $r(t) =\lt -1-2t,2+3t,-2+3t \gt$ and $ \\0 \leq t \leq 1$
Now, the parametric equations are:
$x=-1-2t\\ y= 2+3t; \\z=-2+3t ; \\0 \leq t \leq 1$
