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

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SUMMARY

The discussion clarifies the usage of the equations $r = r_0 + tv$ and $r(t) = (1-t)r_0 + tr_1$. The first equation is employed for a line originating from point $r_0$ in the direction of vector $v$, while the second equation represents a line segment connecting points $r_0$ and $r_1$. Both equations can be interconverted, as demonstrated by rewriting the second equation to match the form of the first. This interchangeability allows for flexibility in choosing which equation to use based on the context of the problem.

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

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