I was trying to find the distance between a point and a 3D line with parametric equations.
On the web, I found a video detailling the steps. https://www.youtube.com/watch?v=9wznbg_aKOo
At 2:20, the person says that the dot-product of any point on the plane and its normal will always be equal.
I can't think of any proof for that! I could visualize it by visualizing the plane y=5 and the n trying dot-products with some points on the plane. It does work!
I can't find a way to prove it! Help!
Best Answer
Let $p_1, p_2$ be two points on the plane that has normal vector $n$. Then the $v=p_1-p_2$ is perpendicular to $n$. So $p_1 \cdot n = p_2\cdot n + (p_1-p_2) \cdot n = p_2\cdot n + 0$.