# Thread: Help expressing distance in the following form...

1. ## Help expressing distance in the following form...

Show that in 3-space the distance $d$ from a point $p$ to the line $l$ through points $a$ and $b$ can be expressed as $d=\frac{|ap\times{ab}|}{|ab|}$.

I was asked this question by my girlfriend who is taking calculus three this summer. I took the class a couple semesters ago and cannot remember this stuff for the life of me. I'm refreshing now, so maybe I will get given a little more time, but for now I am pretty stuck.

Any help would be appreciated.

Thanks

2. Originally Posted by Danneedshelp
Show that in 3-space the distance $d$ from a point $p$ to the line $l$ through points $a$ and $b$ can be expressed as $d=\frac{|ap\times{ab}|}{|ab|}$.
Recall that $\sin (\phi ) = \frac{{\left\| {\overrightarrow {AP} \times \overrightarrow {AB} } \right\|}}{{\left\| {\overrightarrow {AP} } \right\|\left\| {\overrightarrow {AB} } \right\|}}$ where $\phi$ is the angle between $\overrightarrow {AP}~\&~\overrightarrow {AB}$.
The distance is the length of a leg of a right triangle opposite $\phi$.