Simple Nature - Light and Matter

14 deg, and it points along an axis midway between the x and yaxes.4.3.2 Angular momentum in three dimensionsThe vector cross productIn order to expand our system of three-dimensional kinematics toinclude dynamics, we will have to generalize equations like v t = ωr,τ = rF sin θ rF , and L = rp sin θ rp , each of which involves threequantities that we have either already defined as vectors or that wewant to redefine as vectors. Although the first one appears to differfrom the others in its form, it could just as well be rewritten asv t = ωr sin θ ωr , since θ ωr = 90 ◦ , and sin θ ωr = 1.It thus appears that we have discovered something general aboutthe physically useful way to relate three vectors in a multiplicativeway: the magnitude of the result always seems to be proportional tothe product of the magnitudes of the two vectors being “multiplied,”and also to the sine of the angle between them.Is this pattern just an accident? Actually the sine factor hasa very important physical property: it goes to zero when the twovectors are parallel. This is a Good Thing. The generalization ofangular momentum into a three-dimensional vector, for example, ispresumably going to describe not just the clockwise or counterclockwisenature of the motion but also from which direction we wouldhave to view the motion so that it was clockwise or counterclockwise.(A clock’s hands go counterclockwise as seen from behind theclock, and don’t rotate at all as seen from above or to the side.) Nowsuppose a particle is moving directly away from the origin, so thatits r and p vectors are parallel. It is not going around the originfrom any point of view, so its angular momentum vector had betterbe zero.Thinking in a slightly more abstract way, we would expect theangular momentum vector to point perpendicular to the plane ofmotion, just as the angular velocity vector points perpendicular tothe plane of motion. The plane of motion is the plane containingboth r and p, if we place the two vectors tail-to-tail. But if r andp are parallel and are placed tail-to-tail, then there are infinitelymany planes containing them both. To pick one of these planes inpreference to the others would violate the symmetry of space, sincethey should all be equally good. Thus the zero-if-parallel propertyis a necessary consequence of the underlying symmetry of the lawsof physics.The following definition of a kind of vector multiplication is consistentwith everything we’ve seen so far, and on p. 912 we’ll provethat the definition is unique, i.e., if we believe in the symmetry ofspace, it is essentially the only way of defining the multiplication of280 Chapter 4 Conservation of Angular Momentum