Although angular velocity and angular acceleration can be treated as vectors, the angular displacement ???, 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 ???1. Then rotate it by 90° about a vertical axis so that the left edge comes toward you. Call this angular displacement ???2. The spine of the book should now face you, with the writing on it oriented so quit 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 ???1 + ???2 equal ???2 + ???1? Now repeat this experiment but this line with an angle of 1° rather than 90°. Do you think that the infinitesimal displacement obeys the commutative law addition and hence qualifies as a vector? If so, how is the direction of related to the direction of ?
Solution 8DQ Step 1: Rotation in 3-dimensions is not commutative. Which means that, you rotate a book about some axis by /2, and again rotated by some different axis by /2 and end up somewhere. Then you do the reverse by rotating in the reverse direction by keeping track of the axis by which you are rotating 1st and 2nd. By intuition you should reach at the point from where you have started. But you would end up in a different place. So, rotation is not commutative in 3 or higher dimensions. Step 2: Let’s consider an infinitesimal rotation about some arbitrary axis by some small angle . We always talk about the rotation of a vector, whose magnitude does not change by a rotation. There are many such things like the dot products of two vectors which remain invariant by the rotation of the coordinates or by the rotation of the vectors itself. The rotation can be expressed by an operator or a matrix which is denoted as R. The matrix must obey certain rules as, T 1 R = R (orthogonal) det R = ± 1 Where there are significance of +1 and -1 by the group theory.