Are Angular Quantities Vectors? — Question 3

Question 3

Finite rotations fail the ordinary vector addition test

A body-fixed arrow initially points along $+z$. Apply right-handed $90^\circ$ rotations about fixed laboratory axes. Sequence A is first about $x$, then about $y$; sequence B uses the reverse order.

1. Find the final arrow direction for each sequence.

2. Explain what this shows about treating arbitrary finite angular displacements as ordinary vectors.

Question 3 - Solution

Finite rotations fail the ordinary vector addition test

(a) Derivation and calculation

Act on the initial unit vector in the stated order: $$\hat z\xrightarrow{R_x(\pi/2)}-\hat y \xrightarrow{R_y(\pi/2)}-\hat y,\qquad\boxed{\hat u_A=-\hat y}.$$ $$\hat z\xrightarrow{R_y(\pi/2)}\hat x \xrightarrow{R_x(\pi/2)}\hat x,\qquad\boxed{\hat u_B=\hat x}.$$

(b) Result and interpretation

A rotation about an axis leaves a vector along that axis unchanged. The two compositions therefore satisfy $$R_yR_x\hat z=-\hat y\ne\hat x=R_xR_y\hat z.$$ Ordinary addition obeys $\vec a+\vec b=\vec b+\vec a$; arbitrary finite rotations fail this test and cannot be added as ordinary vectors.

Physical check

The final directions are perpendicular, making the order dependence explicit.