In geometry, two lines are said to be skew lines if they do not intersect but are not parallel. Geometry (from the Greek words Ge = earth and metro = measure) is the branch of mathematics first introduced by Thales (circa 624-547 BC) dealing with spatial relationships. ...

Skew lines only exist in three or more dimensions; any two lines in the plane which are not parallel must intersect at some point. In fact, two lines are skew lines if and only if they do not lie in a single plane together. This means that if each line is defined by two points, these four points must not be coplanar; put another way, they must be the vertices of a tetrahedron of nonzero volume. Any three of them will still be coplanar, since three points define a plane, but no three points will be colinear, since this would make all four points coplanar. A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ...

The signed volume of a tetrahedron with vertices v₁=(x₁,y₁,z₁), v₂=(x₂,y₂,z₂), v₃=(x₃,y₃,z₃), and v₄=(x₄,y₄,z₄) can be found using the determinant below. Since this volume must be nonzero for the four points to define skew lines, we can identify skew lines as those satisfying: In linear algebra, the determinant is a function that associates a scalar det(A) to every square matrix A. The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear transformation. ...

Using the definition of the cross product and dot product, we can express this as the equivalent vector inequality: In mathematics, the cross product is a binary operation on vectors in vector space. ... In mathematics, the dot product (also known as the scalar product and the inner product) is a sesquilinear function (·) : V × V → F, where V is a vector space over the field F, having some further properties. ...

Two randomly chosen lines will almost surely be skew lines, because given any three points defining a plane, that plane has zero volume, and so there is zero probability that a randomly-chosen fourth point will fall on it (the probability that the first three points will not define a plane is also zero). Similarly, a very small perturbation of two parallel or intersecting lines will almost surely turn them into skew lines. In this sense, skew lines are the "usual" case, and parallel or intersecting lines are more unusual degenerate cases. In mathematics—specifically, in probability theory—the phrase almost surely is a subtle, precise way to say that something is certain except for cases that almost never happen, though still possible. ...

External links

• Mathworld: Skew Lines (http://mathworld.wolfram.com/SkewLines.html)
• Finding the shortest distance between 2 skew lines (http://www.netcomuk.co.uk/~jenolive/skew.html)