Slope between two points - Vectors

Question

I have two vector points, $\langle 1,3,7 \rangle$ and $\langle 23,-5,10 \rangle$. Point $1$ is at $t=13$ and Point $2$ is at $t=81$. I know how to find the slope between two vector points when $t=0$ and $t=1$ (namely $\langle 1,3,7 \rangle + t\langle 22,-8,3 \rangle$), how do you do it with times like this? Thank you.

Answer 1

The vector between point 1 and point 2 is $\langle23,-5,10\rangle - \langle1,3,7,\rangle = \langle22,-8,3\rangle$.
The time between point 1 and point 2 is $t=81-13 = 68$.
This means that over $68$ units of time, the point traveled over $\langle22,-8,3\rangle$.
It follows so that over $1$ unit of time, the point will travel $\frac1{68}\langle22,-8,3\rangle = \langle\frac{11}{34},-\frac{2}{17},\frac3{68}\rangle$.

And that is your "slope", aka "direction vector!"

Your line, of course, will have the equation $\langle1,3,7\rangle + t\langle\frac{11}{34},-\frac{2}{17},\frac3{68}\rangle$.

Answer 2

To generate a vector equation of the line that hits the target points at the targeted time, then we need to make a small adjustment to $t$

$\langle 1,3,7 \rangle + \frac {t-13}{81-13}\langle 22,-8,3 \rangle$