Although angular velocity and angular acceleration can be treated as vectors, the angular displacement , despite having

Although angular velocity and angular acceleration can be treated as vectors, the angular displacement θ, despite having a magnitude and a direction, cannot. This is because does not follow the commutative law of vector addition (Eq. 1.3). Prove this to yourself in the following way: Lay your physics textbook flat on the desk in front of you with the cover side up so you can read the writing on it. Rotate it through 90° about a horizontal axis so that the farthest edge comes toward you. Call this angular displacement θ₁. Then rotate it by 90º about a vertical axis so that the left edge comes toward you. Call this angular displacement θ_2. The spine of the book should now face you, with the writing on it oriented so that you can read it. Now start over again but carry out the two rotations in the reverse order. Do you get a different result? That is, does θ₁ + θ₂ equal θ₂ + θ₁? Now repeat this experiment but this time with an angle of 1º rather than 90º. Do you think that the infinitesimal displacement dθ obeys the commutative law of addition and hence qualifies as a vector? If so, how is the direction of dθ related to the direction of ω?