University Physics I - Classical Mechanics, 2019

198 CHAPTER 9. ROTATIONAL DYNAMICS
A × B
A
B
A
B
B × A
Figure 9.4: The “right-hand rule” to determine the direction of the cross product. Line up the first vector
with the fingers, and the second vector with the flat of the hand, and the thumb will point in the correct
direction. In the first drawing, we are looking at the plane formed by ⃗ A and ⃗ B from above; in the second
drawing, we are looking at the plane from below, and calculating ⃗ B × ⃗ A.
It follows from Eq. (9.10) that the cross-product of any vector with itself must be zero. In fact,
according to Eq. (9.9), the cross product of any two vectors that are parallel to each other is zero,
since in that case θ = 0, and sin 0 = 0. In this respect, the cross product is the opposite of the
dot product that we introduced in Chapter 7: it is maximum when the vectors being multiplied
are orthogonal, and zero when they are parallel. (And, of course, the result of ⃗ A × ⃗ B is a vector,
whereas ⃗ A · ⃗B is a scalar.)
Besides not being commutative, the cross product also does not have the associative property of
ordinary multiplication: ⃗ A ×( ⃗ B × ⃗ C) is different from ( ⃗ A× ⃗ B)× ⃗ C. You can see this easily from the
fact that, if ⃗ A = ⃗ B, the second expression will be zero, but the first one generally will be nonzero
(since ⃗A × ⃗C is not parallel, but rather perpendicular to ⃗A).
In spite of these oddities, the cross product is extremely useful in physics. We will use it to define
the angular momentum vector ⃗ L of a particle, relative to a point O, as follows:
⃗L = ⃗r × ⃗p = m⃗r × ⃗v (9.11)
where ⃗r is the position vector of the particle, relative to the point O. This definition gives us
a constant vector for a particle moving on a straight line, as discussed in the previous section:
the magnitude of L, ⃗ according to Eq. (9.9) will be mrv sin θ, which, as shown in Fig. 9.1, does
not change as the particle moves. As for the direction, it is always perpendicular to the plane
containing ⃗r and ⃗v (the plane of the paper, in Fig. 9.1), and if you imagine moving ⃗v to point O,
keeping it parallel to itself, and apply the right-hand rule, you will see that L ⃗ in Fig. 9.1 should
point into the plane of the paper at all times.
To see how the definition (9.11) works for a particle moving in a circle, consider again the situation
showninFigure8.6 in the previous chapter, but now extend it to three dimensions, as in Fig. 9.5,
on the next page. It is straightforward to verify that, for the direction of motion shown, the cross