Coplanarity

In three-dimensional space, two linearly 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. This leads to the following coplanarity test using a scalar triple product: Four distinct points, , are coplanar if and only if, :[(x_2 - x_1) \times (x_4 - x_1)] \cdot (x_3 - x_1) = 0. which is also equivalent to :(x_2 - x_1) \cdot [(x_4 - x_1) \times (x_3 - x_1)] = 0. If three vectors are coplanar, then if (i.e., and are orthogonal) then :(\mathbf{c}\cdot\mathbf{\hat a})\mathbf{\hat a} + (\mathbf{c}\cdot\mathbf{\hat b})\mathbf{\hat b} = \mathbf{c}, where denotes the unit vector in the direction of . That is, the vector projections of on and on add to give the original . ax+by+cz+d=0, where is the normal vector of the plane. The value can be computed by substituting one of the points and then solving. If is the same for all subsets of three points, then the planes are the same. One advantage of this technique is that it can work in higher dimensional spaces. For example, suppose one wants to compute the dihedral angle between two -dimensional hyperplanes defined by points in ()-dimensional space. If , then there are an infinite number of normal vectors for each hyperplane, so the angle between two of them is not necessarily the dihedral angle. However, if the Gram–Schmidt process is used, using the same initial vector in both cases, then the angle between the two normal vectors will be minimal, and therefore will be the dihedral angle between the hyperplanes. --> ==Coplanarity of points in n dimensions whose coordinates are given==

Since three or fewer points are always coplanar, the problem of determining when a set of points is coplanar is generally of interest only when there are at least four points involved. In the case that there are exactly four points, several ad hoc methods can be employed, but a general method that works for any number of points uses vector methods and the property that a plane is determined by two linearly independent vectors. In an -dimensional space where , a set of points \{p_0,\ p_1,\ \dots,\ p_{k-1}\} are coplanar if and only if the matrix of their relative differences, that is, the matrix whose columns (or rows) are the vectors \overrightarrow{p_0 p_1},\ \overrightarrow{p_0 p_2},\ \dots,\ \overrightarrow{p_0 p_{k-1}} is of rank 2 or less. For example, given four points :\begin{align} X &= (x_1, x_2, \dots, x_n), \\ Y &= (y_1, y_2, \dots, y_n), \\ Z &= (z_1, z_2, \dots, z_n), \\ W &= (w_1, w_2, \dots, w_n), \end{align} if the matrix :\begin{bmatrix} x_1 - w_1 & x_2 - w_2 & \dots & x_n - w_n \\ y_1 - w_1 & y_2 - w_2 & \dots & y_n - w_n \\ z_1 - w_1 & z_2 - w_2 & \dots & z_n - w_n \\ \end{bmatrix} is of rank 2 or less, the four points are coplanar. In the special case of a plane that contains the origin, the property can be simplified in the following way: A set of points and the origin are coplanar if and only if the matrix of the coordinates of the points is of rank 2 or less. ==Geometric shapes==

A skew polygon is a polygon whose vertices are not coplanar. Such a polygon must have at least four vertices; there are no skew triangles. A polyhedron that has positive volume has vertices that are not all coplanar. ==See also==