Answer
$r(t) =\lt -1-2t,2+3t,-2+3t \gt$ and $ \\0 \leq t \leq 1$
$x=-1-2t$ and $ y= 2+3t $and $z=-2+3t$ ; \\0 \leq t \leq 1$
Work Step by Step
The General vector line equation for the given two points is defined as:
$r(t)=(1-t) r_0+t \times r_1$
Now, we have $r(t) =\lt -1, 2,-2 \gt +t \lt -3, 5,1 \gt$
$\implies \lt -1+t,2-2 \times t, -2+2 \times t \gt + \lt -(3) \times t,5 \times t,t \gt$
$\implies \lt -1-2t,2+3t,-2+3t \gt$
Hence,
$r(t) =\lt -1-2t,2+3t,-2+3t \gt$ and $ \\0 \leq t \leq 1$
Also, the parametric equations are as follows:
$x=-1-2t$ and $ y= 2+3t $and $z=-2+3t$ ;$0 \leq t \leq 1$