Are angular quantities vector? ... It is not easy to get used to representing angular quantities as vectors. We instinctively expect that something should be moving along the direction of a vector. That is not the case here. Instead, something(the rigid body) is rotating around the direction of the vector. In the world of pure rotation,a vector defines an axis of rotation, not a direction in which something moves.
This is what my book (authored by Halliday, Walker, Resnick) says. But they mentioned with a caution that
Angular displacements cannot be treated as vectors. ....to be represented as a vector, a quantity must also obey the rules of vector addition..... Angular displacements fail this test.
Why do they fail in this test? What is the reason?
Answer
Set your copy of Halliday, Walker & Resnick on the table so the front cover is parallel to the table and visible to you. Now your right hand flat on the book with your thumb and forefinger forming a ninety degree angle. Orient your hand so your thumb is parallel to the spine of the book and pointing toward the top edge. Your forefinger should be parallel to the lines of text on the cover, pointing to the right.
This makes a nice basis for a book-based coordinate system. Your thumb points along the x-hat axis, your forefinger along the y-hat axis. To complete a right-hand system, the z-hat axis points into the book.
Pick the book up and make a +90 degree rotation about the x-hat axis. The spine of the book should be horizontal and facing up. Now make a +90 degree rotation about the y-hat axis (as rotated by that first rotation). You should be looking at the front cover of the book but oriented vertically with text flowing toward the ground.
If you start all over again but reverse the order of operations, +90 degree rotation about the y-hat axis followed by a +90 degree rotation about the x-hat axis, you should be looking at the spine of the book rather than the front cover. The front cover is oriented vertically, but with text running parallel to the ground. You can put your book back in the bookcase.
Rotation in three dimensional space and higher is not commutative (rotation A followed by rotation B is not necessarily the same as rotation B followed by rotation A).
Another hint that there's something different between rotation and translation is the number of parameters needed to describe the two in some N-dimensional space. Translation in N dimensional space obviously needs N parameters. Lines have one degree of freedom, planes, two, three dimensional space three, and so on. Lines don't rotate. There are no rotational degrees of freedom in one dimensional space. Rotation does make sense in two dimensional space, where a single scalar (one rotational degree of freedom) completely describes rotations. Three dimensional space has three degrees of freedom. Four dimensional space? It has six. Three dimensional space is the only space for which the number of rotational degrees of freedom and number of translational degrees of freedom are equal to one another.
This unique characteristic of three dimensional space is why you can treat angular velocity as a vector. The vector cross product (something else that is unique to three dimensional space) means introductory students can be taught about rotations without having to learn about Lie theory or abstract algebras. That wouldn't be the case for students who live in a universe with four spatial dimensions.
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