Answer
$r(t) =\lt a+t(u-a),b+t(v-b),c+t(w-c) \gt$
$x=a+t(u-a) \\ y= b+t(v-b); \\z=c+t(w-c) ; \\0 \leq t \leq 1$
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:
$r(t)=(1-t) r_0+tr_1=(1-t) \lt a,b,c \gt +t \lt u,v,w \gt$
or, $=\lt a-at,b-bt, c-ct \gt + \lt ut,vt,wt\gt$
or, $= \lt a+t(u-a),b+t(v-b),c+t(w-c) \gt$
Thus, we have $r(t) =\lt a+t(u-a),b+t(v-b),c+t(w-c) \gt$
Now, the parametric equations are:
$x=a+t(u-a) \\ y= b+t(v-b); \\z=c+t(w-c) ; \\0 \leq t \leq 1$
