I am reading "Berkeley Physics Course Mechanics Volume 1 Second Edition" by Charles Kittel et al..
To be a vector a quantity must satisfy two conditions:
- It must satisfy the parallelogram law of addition.
- It must have a magnitude and a direction independent of the choice of coordinate system.
Finite Rotations Are Not Vectors
Not all quantities that have magnitude and direction are necessarily vectors. For example, a rotation of a rigid body about a particular axis fixed in space has a magnitude (the angle of rotation) and a direction (the direction of the axis). But two such rotations do not combine according to the vector law of addition, unless the angles of rotation are infinitesimally small. This is easily seen when the two axes are perpendicular to each other and the rotations are by $\frac{\pi}{2}$ rad $(90^\circ)$. Consider the object (a book) in Fig. 2.8a. The rotation (1) leaves it as in Fig. 2.8b, and a subsequent rotation (2) about another axis leaves the object as in Fig. 2.8c. But if to the object as originally oriented (Fig. 2.8d) we apply first the rotation (2), (Fig. 2.8e) and then the rotation (1), the object ends up as shown in Fig. 2.8f. The orientation in the sixth figure is not the same as in the third. Obviously the commutative law of addition is not satisfied by these rotations. Despite the fact that they have a magnitude and a direction, finite rotations cannot be represented as vectors.
The authors showed finite rotations are not vectors because finite rotations don't satisfy the axioms of a vector space. (the commutative law.)
This book is a very famous book, but I doubt this example is a good example.
For example,
Suppose $v_1,v_2$ are Euclidean vectors.
We define $v_1+v_2:=v_1-v_2$. ("$-$" is the usual subtraction operator.)
Then, obviously this addition doesn't satisfy the axioms of a vector space.
We can easily make examples like this in which the sum of vectors doesn't satisfy the axioms of a vector space.
Is the example in the book a good example?