Rotation Matrix

linear-algebra

Definition

Rotation Matrix

A rotation matrix is an orthogonal matrix with determinant .

It represents a linear map that preserves lengths, angles, and orientation.

Examples

Rotation in

A rotation in turns every vector about the origin by a fixed angle .
It does not change the length of the vector; it only changes its direction.

Derivation

A rotation is determined once we know what it does to the standard basis vectors and .

Since is a rotation by , the image of is the unit vector making angle with the positive -axis. Hence

The vector is orthogonal to . A rotation preserves angles and orientation, so must be the unit vector obtained by rotating by another . Therefore

Now write an arbitrary vector as

By linearity,

Substituting the two basis images gives

The minus sign is therefore forced: it is the -component of , and points to the left.

In , the rotation by an angle is given by

If , then

The columns of are the images of the standard basis vectors and under the rotation.

Special case: For ,

Mirroring on the -Axis

Projection onto the First Median