So I have two points in 3D space, lets call them $p_1=(2,1,-1)$ and $p_2=(3,2,-2)\ $. This is all the information I have about these points.
If I wish to extend this line to a $p_3$, how would I do this?
My current workings out have led me to this:
If $t = 2$:
$p_3 = p_1 + t\cdot(p_2 - p_1)$
$p_3 = (2,1,-1) + 2 \cdot ((3,2,-2)-(2,1,-1)$
$p_3 = (4,3,-7)\ $.
But to me that $-7$ sticks out like a sore thumb, there is surely no way that can be correct. Is there any mistakes? Thank you.
What you do to extend a line in 2D or 3D is qualitatively the same.
Just leave $t$ as it is, so you know all points $p_3$ as a function of $t$.
$ p_3 = p_1 + t\cdot(p_2 - p_1) $
$ p_3 = (2,1,-1) + t \cdot ((3,2,-2)-(2,1,-1) $
$ p_3 = (2,1,-1) + t \cdot ((1,1,-1) $
For t = 1, $ p_3 = (3,2,-2)$ ;
For t = 2, $ p_3 = (4,3,-3)$ ;
etc.