 For example, the three-dimensional object physics calls angular velocity is a differential rotation, thus a vector in the Lie algebra tangent to .Given , with being a unit vector, the correct skew-symmetric matrix form of is

The exponential of this is the orthogonal matrix for rotation around axis by angle ; setting , ,

Numerical analysis takes advantage of many of the properties of orthogonal matrices for numerical linear algebra, and they arise naturally.For example, it is often desirable to compute an orthonormal basis for a space, or an orthogonal change of bases; both take the form of orthogonal matrices.