[Math] Find a vector parametric equation $\vec r(t)$ for the line through the points $P\equiv(1,0,−4)$ and$ Q\equiv(3,−3,1)$

Question

Find a vector parametric equation $\vec r(t)$ for the line through the points $P\equiv(1,0,−4)$ and $Q\equiv(3,−3,1)$ for each of the given conditions on the parameter $t$

If $\vec r(3)=P$ and $\vec r(7)=Q$

I get how to do it then I have $r(0)$ but im not sure what to do with the $r(3)$

Answer 1

$$r(t) = \frac{(7-t)}{(7-3)}(1,0,-4) + \frac{(t-3)}{(7-3)}(3,-3,1)$$ will do.