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

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

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

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