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

  1. #1
    Senior Member Danneedshelp's Avatar
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    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
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    Quote Originally Posted by Danneedshelp View Post
    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 $\displaystyle \phi$.
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