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Testing for skewness

If each line in a pair of skew lines is defined by two points that it passes through, then these four points must not be coplanar, so they must be the vertices of a tetrahedron of nonzero volume. Conversely, any two pairs of points defining a tetrahedron of nonzero volume also define a pair of skew lines. Therefore, a test of whether two pairs of points define skew lines is to apply the formula for the volume of a tetrahedron in terms of its four vertices. Denoting one point as the 1×3 vector a whose three elements are the point's three coordinate values, and likewise denoting b, c, and d for the other points, we can check if the line through a and b is skew to the line through c and d by seeing if the tetrahedron volume formula gives a non-zero result: