Discuss whether or not , angular displacement is a vector quantity ?`

Explanation

Angular displacement is a vector quantity provided andle ` theta` is small because the commutative law of vector addirion for large anle is not valid where as for small angles, the law is valid i.e. ` theta_(1) + that_(2) != theta_(2) + theta_(1)`, but ` Delta theta_(1) + Delta_(2) =Delta _(1)`. The same is explained below.
Consider a book a with front page (F) lying in the plane of paper. It can be rotated about two mutually perpendicular azes ` 1 and 2` as shown in Fig. 1 (d) .34(a) (i) Let the book be rotated throught an angle ` theta_(1)` (=90^(@))` in anticleocikwise dirction about axis ` 1`. It takes the position as shown in fig. 2 (d).34 (b). On further rotation the book through an anlgle ` theta+(2) (=90^(@)) in anticlockwese direction about axis `2` the book ocupies the position as shown in ` Fig.2 (d) 34. (c ) .
(ii) Form the initial position of bok , [ Fig.2 (d). 34 (a)], if the book is rotated through ` theta_(1) (=90^(@))` in the anttclockwise direction first about the axis ``2`, it occupties the position as shown in Fig. 2 9d).34 (d). On further rotating the book through `theta_(2) 9=90^(@)) in clockwise direction about axis `1`the book occupies the position shown in Fig. 2 (d). 34 (e).
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For above we note that the final positions of the book shown in Fig. 2 (d) .34 (c ) and Fig. 2 (d) .34 (e) are not the same. Hence
` theta_(1) + theta_(2)!= theta_(2) + theta_(1)`.
This shows that ` theta_(1) and theta_(2)` not vectors as they do not obey the commutative law of vector addition.
If the book is rotated through a smaller angle ` Delta theta _(1) and Delta theta_(2)` (say 2^(@) or 3^(@), the final positions of the book in the two cses discussed above would almost be the samd.As `Delta theta rarr 0`, the final positions of the bookwill become indistinguishable, thence
` Delta theta_(1) + Delta theta_(2) =Delta theta_(2) + Delta theta_(1)`,