Show that the angular velocity of a body, in terms of Euler angles , , and ,

Show that the angular velocity of a body, in terms of Euler angles ϕ, θ, and ψ, can be expressed as ω =(ϕ̇ sin θ sin ψ + θ̇ cos ψ)i + (ϕ̇ sin θ cos ψ - θ̇ sin ψ)j + (ϕ̇ cos θ + ψ̇ )k, where i, j, and k are directed along the x, y, z axes as shown in Fig. 21-15d.