When finding the equation of a line through pairs of points in space, does the orientation of the line matter?
For instance, say we have the two points $(3, -2, 4)$ and $(-5, 7, 1)$. The vector equation of a line is $x = tw + v$, where $w$ and $v$ are vectors, $t$ is a real number, and $x$ is some point on the line.
There are two commonly asked questions:
- Find the endpoint of the vector emanating from the origin and having
the same length and direction as the vector beginning at $A$ and
terminating at $B$. - Find the equations of the lines through the following pairs of
points in space.
My understanding is that, in the case of 1, the order that you manipulate the vectors in matters, since the direction matters (we want a vector in a certain direction – not a line). However, in the case of 2, the order that you manipulate the vectors in does not matter, since a line has no direction. Is this understanding correct?
Returning to the example above, let's assume that we are trying to find the equations of the lines through the pairs of points in space. In this case, there are 4 equivalently correct solutions for the line:
$(-5, 7, 1) – (3, -2, 4) = (-8, 9, -3)$ or $(3, -2, 4) – (-5, 7, 1) = (8, -9, 3)$
$x = tw + v$
- $x = t(-8, 9, -3) + (-5, 7, 1)$
- $x = t(-8, 9, -3) + (3, -2, 4)$
- $x = t(8, -9, 3) + (-5, 7, 1)$
- $x = t(8, -9, 3) + (3, -2, 4)$
Would it be correct to say that, since these are lines (direction does not matter), all 4 are equivalent?
I would greatly appreciate it if the members of MSE could please take the time to review my reasoning.
Thank you.
Best Answer
Yes. In general there are infinitely many distinct parametric equations defining the same curve. For a line, one can multiply the parameter $t$ by any non-zero constant to get another parametric equation.