# Thread: vector proof using cross product

1. ## vector proof using cross product

Let A and B be 2 points on a line and P be a point off the line. Prove that the shortest distance from P to AB is given by the following equation:

$\displaystyle d=\frac{\parallel\vec{AP}\times\vec{AB}\parallel}{ \parallel\vec{AB}\parallel}$

I've tried to solve for d using the Pythagorean theorem and the projection of $\displaystyle \vec{AP}$ onto $\displaystyle \vec{AB}$ for one side and $\displaystyle \vec{AP}$ and d for the other sides, but I'm stumped.

I can't relate it to the cross product. Can anyone help? Thanks!

2. Originally Posted by duaneg37
Let A and B be 2 points on a line and P be a point off the line. Prove that the shortest distance from P to AB is given by the following equation:

$\displaystyle d=\frac{\parallel\vec{AP}\times\vec{AB}\parallel}{ \parallel\vec{AB}\parallel}$
You need to realize that $\displaystyle \left\|\vec{AP}\times\vec{AB}\right\|=\left\|\vec{ AB} \right\|\left\|\vec {AP} \right\|\sin(\theta)$ where $\displaystyle \theta$ is the angle between the vectors.

3. I feel stupid for not remembering that! It's plain as day now. Thank you so much!