Answer
$r(t) =\lt a+t(u-a),b+t(v-b),c+t(w-c) \gt$;
$x=a+t(u-a)$ and $y= b+t(v-b)$ and $z=c+t(w-c)$ and $0 \leq t \leq 1$
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
$r(t)=(1-t) \times \lt a,b,c \gt +t \times \lt u,v,w \gt$
$\implies \lt a-at,b-bt, c-ct \gt + \lt ut,vt,wt\gt$
$\implies \lt a+t(u-a),b+t(v-b),c+t(w-c) \gt$
Answer: $r(t) =\lt a+t(u-a),b+t(v-b),c+t(w-c) \gt$
Hence, the parametric equations are as follows:
$x=a+t(u-a)$ and $y= b+t(v-b)$ and $z=c+t(w-c)$ and $0 \leq t \leq 1$
