TITLE: Angle between two straight lines
QUESTION [5 upvotes]: Two straight lines are given:
$$
\left(
\begin{array}{c}
1\\
1\\
4
\end{array}
\right) + t  \left(
\begin{array}{c}
4\\
1\\
1
\end{array}
\right)
$$
and
$$
\left(
\begin{array}{c}
5\\
5\\
2
\end{array}
\right) + s  \left(
\begin{array}{c}
0\\
1\\
-1
\end{array}
\right)
$$
I want to calculate the angle between them (which should be simple, arccos of the scalar product over the product of their norms). Since the angle should not change no matter which $s$ and $t$ we choose, I took arbitrary parameters $t = 2$ and $s = 1$ and calculated first just the scalar product of $(9,3,6)$ and $(5,6,1)$ - that is clearly non-zero, though the solutions state that the angle between the lines should be 90 degrees - what am I doing wrong?
Thanks

REPLY [4 votes]: So the problem is that once you fix $s$ and $t$, you are fixing two points on the line, and the inner product you get over the two points are related to the angle of the vectors starting from origin $(0,0)$ to those two points. Thus is not correct.
You want to take the inner product of the vector that is of the same direct of the line. And in this case it should be $(4,1,1)$ and $(0,1,-1)$
Intuitively, the lines are defined by one point and a direction, and $t$, $s$ just give points the freedom to move along the direction, which will conform a line.