Help expressing distance in the following form...

**Danneedshelp** · June 14th 2009, 07:44 PM

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

**Plato** · June 15th 2009, 03:09 AM

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 $\phi$ .

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