A Classical Approach of Newtonian Mechanics

8 ROTATIONAL MOTION 8.3 Is rotation a vector?

direction of rotation vector

Figure 68: The right-hand grip rule.

The rotation “vector” φ now has a well-defined magnitude and direction. But,

is this quantity really a vector? This may seem like a strange question to ask,

but it turns out that not all quantities which have well-defined magnitudes and

directions are necessarily vectors. Let us review some properties of vectors. If a

and b are two general vectors, then it is certainly the case that

a + b = b + a. (8.7)

In other words, the addition of vectors is necessarily commutative (i.e., it is in-

dependent of the order of addition). Is this true for “vector” rotations, as we

have just defined them? Figure 69 shows the effect of applying two successive

90 ◦ rotations—one about the x-axis, and the other about the z-axis—to a six-

sided die. In the left-hand case, the z-rotation is applied before the x-rotation,

and vice versa in the right-hand case. It can be seen that the die ends up in two
completely different states. Clearly, the z-rotation plus the x-rotation does not

equal the x-rotation plus the z-rotation. This non-commutative algebra cannot be

represented by vectors. We conclude that, although rotations have well-defined

magnitudes and directions, they are not, in general, vector quantities.

There is a direct analogy between rotation and motion over the Earth’s surface.

After all, the motion of a pointer along the Earth’s equator from longitude 0 ◦W to

sense of rotation