Rotation Matrix -- from Wolfram MathWorld

When discussing a rotation, there are two possible conventions: rotation of the axes, and rotation of the object relative to fixed axes.

In , consider the matrix that rotates a given vector by a counterclockwise angle in a fixed coordinate system. Then

(1)

(2)

This is the convention used by the Wolfram Language command RotationMatrix[theta].

On the other hand, consider the matrix that rotates the coordinate system through a counterclockwise angle . The coordinates of the fixed vector in the rotated coordinate system are now given by a rotation matrix which is the transpose of the fixed-axis matrix and, as can be seen in the above diagram, is equivalent to rotating the vector by a counterclockwise angle of relative to a fixed set of axes, giving

(3)

This is the convention commonly used in textbooks such as Arfken (1985, p. 195).

In , coordinate system rotations of the x-, y-, and z-axes in a counterclockwise direction when looking towards the origin give the matrices

			(4)
			(5)
			(6)

(Goldstein 1980, pp. 146-147 and 608; Arfken 1985, pp. 199-200).

Any rotation can be given as a composition of rotations about three axes (Euler's rotation theorem), and thus can be represented by a matrix operating on a vector,

[x_1^'; x_2^'; x_3^']=[a_(11) a_(12) a_(13); a_(21) a_(22) a_(23); a_(31) a_(32) a_(33)][x_1; x_2; x_3].

(7)

We wish to place conditions on this matrix so that it is consistent with an orthogonal transformation (basically, a rotation or improper rotation).

In a rotation, a vector must keep its original length, so it must be true that

(8)

for , 2, 3, where Einstein summation is being used. Therefore, from the transformation equation,

(9)

This can be rearranged to

			(10)
			(11)
			(12)

In order for this to hold, it must be true that

(13)

for , 2, 3, where is the Kronecker delta. This is known as the orthogonality condition, and it guarantees that

(14)

and

(15)

where is the transpose and is the identity matrix. Equation (15) is the identity which gives the orthogonal matrix its name. Orthogonal matrices have special properties which allow them to be manipulated and identified with particular ease.