An important application of the above theory is to the Euler angles, important in the mechanics of
rotating bodies. Lagrange studied these things back in the 1700’s. To describe the Euler angles
consider the following picture in which x_{1},x_{2} and x_{3} are the usual coordinate axes fixed in
space and the axes labeled with a superscript denote other coordinate axes. Here is the
picture.

PICT

PICT

We obtain ϕ by rotating counter clockwise about the fixed x_{3} axis. Thus this rotation has the
matrix

Next rotate counter clockwise about the x_{1}^{1} axis which results from the first rotation through an
angle of θ. Thus it is desired to rotate counter clockwise through an angle θ about the unit
vector

Therefore, in 8.4, a = cosϕ,b = sinϕ, and c = 0. It follows the matrix of this transformation with
respect to the usual basis is

( 2 2 )
| cos ϕ +sin ϕcosθ cos ϕ2sin ϕ(1−2 cosθ) sinϕ sinθ |
( cosϕsin ϕ(1− cosθ) sin ϕ+ cos ϕcosθ − cosϕ sinθ ) ≡ M2 (ϕ,θ)
− sinϕsinθ cosϕ sinθ cosθ

Finally, we rotate counter clockwise about the positive x_{3}^{2} axis by ψ. The vector in the positive
x_{3}^{1} axis is the same as the vector in the fixed x_{3} axis. Thus the unit vector in the positive
direction of the x_{3}^{2} axis is

and it is desired to rotate counter clockwise through an angle of ψ about this vector. Thus, in this
case,

a = cos2ϕ + sin2ϕ cos θ,b = cosϕsin ϕ(1− cosθ),c = − sin ϕsin θ.

and you could substitute in to the formula of Theorem 8.4 and obtain a matrix which represents
the linear transformation obtained by rotating counter clockwise about the positive x_{3}^{2} axis,
M_{3}

(ϕ,θ,ψ)

. Then what would be the matrix with respect to the usual basis for the linear
transformation which is obtained as a composition of the three just described? By Theorem 8.3.17,
this matrix equals the product of these three,

M3 (ϕ,θ,ψ)M2 (ϕ,θ)M1 (ϕ).

I leave the details to you. There are procedures due to Lagrange which will allow you to write
differential equations for the Euler angles in a rotating body. To give an idea how these angles
apply, consider the following picture.

PICT

This is as far as I will go on this topic. The point is, it is possible to give a systematic
description in terms of matrix multiplication of a very elaborate geometrical description of a
composition of linear transformations. You see from the picture it is possible to describe the
motion of the spinning top shown in terms of these Euler angles.

class=”left” align=”middle”(V,W) As A Vector Space8.4. EIGENVALUES AND
EIGENVECTORS OF LINEAR TRANSFORMATIONS