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Mathematics-Online lexicon:

Distance Point - Line


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The projection $ X$ of a point $ Q$ on a line through $ P$ with direction $ \vec{u}$ satisfies

$\displaystyle \overrightarrow{PX} = t\vec{u},\quad
t = \frac{(\vec{q}-\vec{p})\cdot\vec{u}}{\vert\vec{u}\vert^2}
\,.
$

From this we obtain the distance of the point $ Q$ from the line by

$\displaystyle d =
\frac{\left\vert(\vec{q}-\vec{p})\times\vec{u}\right\vert}
{\vert\vec{u}\vert}\,
.
$

\includegraphics[width=12cm]{a_abstand_punkt_gerade_bild}

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  automatically generated 3/28/2008