Coplanarity

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In geometry, some points in space are coplanar if there is a geometric plane that includes them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. However, a set of four or more distinct points will, in general, not lie in a single plane.

Two lines in three-dimensional space are coplanar if there is a plane that includes them both. This occurs if the lines are parallel, or if they intersect each other.

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

Properties[edit]

In three-dimensional space, two independent vectors with the same initial point determine a plane through that point. Their cross product is a normal vector to that plane and any vector orthogonal to this cross product through the initial point will lie in the plane.[1] This leads to the following coplanarity test. Four distinct points, x1, x2, x3 and x4 are coplanar if and only if,

(x_3 - x_1) \cdot [(x_2 - x_1) \times (x_4 - x_3)] = 0.

If three vectors \mathbf{a}, \mathbf{b} and \mathbf{c} are coplanar, then

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

where \mathbf{\hat a} denotes the unit vector in the direction of \mathbf{a}. That is, the vector projections of \mathbf{c} on \mathbf{a} and \mathbf{c} on \mathbf{b} add to give the original \mathbf{c}.


Coplanarity of points whose coordinates are given[edit]

In coordinate geometry, in n-dimensional space, a set of four or more distinct points are coplanar if and only if the matrix of the coordinates of these points is of rank 2 or less. For example, given four points, W = (w1, w2, ... , wn), X = (x1, x2, ... , xn), Y = (y1, y2, ... , yn), and Z = (z1, z2, ... , zn), if the matrix

\begin{bmatrix}
w_1 & w_2 & \dots & w_n  \\
x_1 & x_2 & \dots & x_n  \\
y_1 & y_2 & \dots & y_n \\
z_1 & z_2 & \dots & z_n
\end{bmatrix}

is of rank 2 or less, the four points are coplanar.

See also[edit]

Collinearity

References[edit]

  1. ^ Swokowski, Earl W. (1983), Calculus with Analytic Geometry (Alternate ed.), Prindle, Weber & Schmidt, p. 647, ISBN 0-87150-341-7 

External links[edit]