MHB Solving for r(t): $r_0+tv$ vs. $ (1-t)r_0+tr_1$

[ineedhelpnow]

how do i know when I am supposed to use this $r=r_0+tv$ and when I am supposed to use $r(t)=(1-t)r_0+tr_1$

 

[I like Serena]

how do i know when I am supposed to use this $r=r_0+tv$ and when I am supposed to use $r(t)=(1-t)r_0+tr_1$

The first is for a line through $r_0$ in the direction of $v$.
The second is for a line through $r_0$ and $r_1$.

Try it with $t=0$ respectively $t=1$ and you'll see! (Smile)

 

[ineedhelpnow]

sometimes I am given two points and what i do is find the vector between the two points and i use the first equation. could i have just used the second instead given the two points?

 

[I like Serena]

sometimes I am given two points and what i do is find the vector between the two points and i use the first equation. could i have just used the second instead given the two points?

Yep.

Actually, it's the same thing.
Note that the second can be rewritten as the first as follows:
$$r(t)=(1-t)r_0 + t\,r_1=r_0 + t(r_1 - r_0)$$

The vector $r_1 - r_0$ is the vector along the line that runs from the first point to the second point.