Figures 4.9 and 4.10 show that the order of finite rotations leads to different results. Use the notation that A(α,1n) where A is a rotation in the direction of 1n through an angle α. Let n1 and n2 be two orthogonal directions.
(a) If x is the position vector of a point on a rigid body, which is then rotated by an angle θ around the origin, show that the new value of x is
x′=(1n⋅x)1n+[x−1n(1n⋅x)]cosθ−1n×xsinθ.
From this, obtain the formula for A(π/2,1n) and derive the two rotations in the figures.
(b) Discuss these two rotations. [*Hint:* The answer will involve a rotation by the angle 32π in a direction (1/3)(1,1,1).]