Parameterize line to correspond to t values

Question

Parameterize the line through P=(−3,−1) and Q=(0,7) so that the points P and Q correspond to the parameter values t=13 and 16.

I have found the direction vector <3,8> and then have plugged in to find the following:

r=<-3,-1> +t<3,8>

x=3t-3 y=8t-1

I am lost as to the next steps to scale this appropriately to come up with the correct equation.

Answer 1

Here is an alternate approach beginning with what you started.

$$r=\langle -3,-1\rangle+t\langle 3,8 \rangle$$

Now this takes us from the first to the second point too quickly, in one unit of $t$ when it should take three units of $t$. So we replace $t$ with $t/3$ to get

$$r=\langle -3,-1\rangle+\frac{t}{3}\langle 3,8 \rangle$$

$$r=\langle -3,-1\rangle+t\left\langle 1,\frac{8}{3} \right\rangle$$

But now, we are arriving at the second point when $t=3$ which is $13$ units of $t$ too soon. So we should delay by $13$ units of $t$ so we arrive when $t=16$.

$$r=\langle -3,-1\rangle+(t-13)\left\langle 1,\frac{8}{3} \right\rangle$$

$$r=\langle -3,-1\rangle+t\left\langle 1,\frac{8}{3} \right\rangle-13\left\langle 1,\frac{8}{3} \right\rangle$$

$$r=\left\langle -16,-\frac{107}{3}\right\rangle+t\left\langle 1,\frac{8}{3} \right\rangle$$

ADDENDUM:

Note that all of this may be summarized in a formula.

If for $t_0<t_1$, $r(t)$ is a linear function satisfying $r(t_0)=\langle x_0,y_0\rangle$ and $r(t_1)=\langle x_1,y_1\rangle$ and if $t_0\le t\le t_1$ then

$$ r(t)=\langle x_0,y_0\rangle+\frac{t-t_0}{t_1-t_0}\langle x_1-x_0,y_1-y_0\rangle $$

Answer 2

Any straight line can be written in parametric equations x= At+ B, y= Ct+ D. We want the point (-3, -1) to correspond to t= 13. That is, we want 13A+ B= -3 and 13C+ D= -1. We want the point (0, 7) to correspond to t= 16 That is, we want 16A+ B= 0 and 16C+ D= 7.

Solve the equations 13A+ B= -3 and 16A+ B= 0 for A and B.
Solve the equations 13C+ D= -1 and 16C+ D= 7 for C and D.