In geometry, a set of points in space is coplanar if the points all lie in a geometric plane. For example, three points are always coplanar; but four points in space are usually not coplanar.

Points can be shown to be coplanar by determining that the scalar product of a vector that is normal to the plane and a vector from any point on the plane to the point being tested is 0.

Distance geometry provides a solution to the problem of determining if a set of points is coplanar, knowing only the distances between them.

Properties

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If three 3-dimensional vectors $\backslash mathbf\{a\},\; \backslash mathbf\{b\}$ and $\backslash mathbf\{c\}$ are coplanar, and $\backslash mathbf\{a\}\backslash cdot\backslash mathbf\{b\}\; =\; 0$, then

$(\backslash mathbf\{c\}\backslash cdot\backslash mathbf\{\backslash hat\; a\})\backslash cdot\backslash mathbf\{\backslash hat\; a\}\; +\; (\backslash mathbf\{c\}\backslash cdot\backslash mathbf\{\backslash hat\; b\})\backslash cdot\backslash mathbf\{\backslash hat\; b\}\; =\; \backslash mathbf\{\backslash hat\; c\},$

where $\backslash mathbf\{\backslash hat\; a\}$ denotes the unit vector in the direction of $\backslash mathbf\{a\}$.

Or, the vector resolutes of $\backslash mathbf\{c\}$ on $\backslash mathbf\{a\}$ and $\backslash mathbf\{c\}$ on $\backslash mathbf\{b\}$ add to give the original $\backslash mathbf\{c\}$.

External link

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Geometry