I've looked for many similar questions but could not find an answer that helps.
I have a point $p$ $(x,y,z)$ on a plane, and a normal to the plane $n$ $(a,b,c)$. I need to find the equation of a line defined by the point $p$ and normal $n$.
Then I would like to know how to find the distance from any arbitrary point $p_a$ $(x_a,y_a,z_a)$ and the line defined above.
A line through a point $p$ which is parallel to the plane normal $n$ (or in other words - the line through a point $p$ normal to the plane with normal $n$) is simply:
$${\bf p} + {\bf n} \cdot t$$ Where $t$ is the line parameter. To find the distance between a third point $q$ and this line, you may want to use the following formula:
$$d = \left|({\bf q}-{\bf p})\times({\bf q}-{\bf p}-{\bf{n}}/|{\bf n}|)\right|$$ Where $\times$ is the cross product.