I have a line defined as $r(t) = a + tb$, where $a$ and $b$ are vectors.
$p$ is a point on the line when $t = t_1$.
How do i obtain a normal vector to the line that passes through $p$?
According to How to find normal vector of line given point normal passes through,
the normal is simply $a + t_1 * b – p$?
But why is that the answer? point $a$ is not even on our normal line?
Best Answer
While $a$ is not (necessarily) a point on the normal line, $r=a+t_1 b$ is. This is because $t_1$ was calculated exactly such that a the normal line would intersect the line you started with at this point. Thus, $r=a+t_1 b - p$ is exactly the vector connecting $p$ and the line orthogonally.
However, in the very special case that $p$ is already on the line, the question is ill-posed, as there is no well-defined direction in which the normal line would point, starting at $p$. In that case, an infinite amount of normal lines would be possible, forming the plane orthogonally intersecting your line at $p$.