Vector Angle

The vector angle between two nonzero real vectors and is the angle defined by

where the numerator is the dot product and the denominators are Euclidean vector norms. Schwarz's inequality ensures that the argument of the inverse cosine lies in . In two or three dimensions, this is the smaller angle between arrows with coincident tails.

The vector angle is 0 for vectors pointing in the same direction, for orthogonal vectors, and for opposite directions. It is undefined if either argument is the zero vector. For example, and have vector angle .

Kahan (2016, p. 15) recommends a half-angle formula that can be written using the two-argument inverse tangent as

where and are unit vectors, and corresponds to ArcTan[x, y] in the Wolfram Language. This form avoids the loss of accuracy of the inverse cosine formula near 0 and . The vector norms must be computed with suitable scaling to avoid floating-point overflow and underflow.

Unlike the smaller line-line angle between unoriented lines, the vector angle retains the directions of its arguments. Reversing one direction replaces by . A dihedral angle can be related to the vector angle between normal vectors to the two planes, with the choice of normals determining which supplementary angle is obtained.

The Wolfram Language computes the vector angle using VectorAngle[u, v]. The related function AngleVector[theta] instead constructs the planar unit vector at a counterclockwise angle from the positive -axis. Its orientation is specified modulo , whereas the real vector angle is restricted to .