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

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Discussion Overview

The discussion revolves around the conditions under which to use two different equations for representing a line in vector form: $r = r_0 + tv$ and $r(t) = (1-t)r_0 + tr_1$. Participants explore the contexts and interpretations of these equations, particularly in relation to points and vectors in geometry.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants suggest that $r = r_0 + tv$ is used for a line through a point $r_0$ in the direction of a vector $v$.
  • Others argue that $r(t) = (1-t)r_0 + tr_1$ represents a line that connects two points, $r_0$ and $r_1$.
  • A participant mentions using the first equation when given two points and questions whether the second equation could also be applicable in that scenario.
  • Another participant confirms that both equations can represent the same line, noting that the second can be rewritten to resemble the first by expressing the direction vector as $r_1 - r_0$.

Areas of Agreement / Disagreement

Participants express some agreement that both equations can describe the same line under certain conditions, but there remains a lack of consensus on when to use each equation specifically.

Contextual Notes

Participants do not fully resolve the conditions under which each equation is preferable, and there are assumptions about the definitions of the vectors and points involved that are not explicitly stated.

ineedhelpnow
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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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