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

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The discussion clarifies when to use the equations $r = r_0 + tv$ and $r(t) = (1-t)r_0 + tr_1$. The first equation represents a line through the point $r_0$ in the direction of vector $v$, while the second describes a line connecting two points, $r_0$ and $r_1$. It is noted that both equations can be related, as the second can be rewritten to resemble the first by substituting $v$ with the vector $r_1 - r_0$. Users can apply either equation depending on the context, with the second being more suitable when given two specific points. Understanding these relationships helps in solving for the parameter $t$ effectively.
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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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