CROSS PRODUCTS (or vector products)

It is useful in physics to define a vector product of vectors A and B to be a vector in a direction perpendicular to the AB plane with magnitude ABsinq . The system described below is standard all over the planet, but not the universe. So if we ever get together with others in the universe, half of them will probably have their cross products going in the opposite direction.

Two ways to do it:

1. The magnitude is the product of their magnitudes times the sine of the angle between them and the direction is the way a screw advances when rotated from the first vector toward the second.

The sine factor provides a measure of the perpendicularity of the two vectors. It varies from 0 to 1 to 0 as the vectors go from parallel to perpendicular to anti-parallel.

2. Define unit vectors in the x, y and z directions respectively: i, j and k. Then A x B = i(A_y B_z - A_z B_y ) + j(....................) + k(......................) where the blanks are filled in according to the cycle xyzxyx... That is, in the j parentheses the first term is
A_z B_x because of A_y B_z in the i parentheses. This is not as much of a mess as it seems, because if you define your xy plane such that A and B are in it, then the i and j components of the cross product vanish and A x B = k(A_x B_y - A_y B_x ).

alt 2. If you know determinants you will recognize the above as the determinant,

Note that if A and B are in the xy plane, AxB is in the z or -z direction. If you are looking at the plane, a clockwise rotation is away from you, and an object spinning that way has its angular momentum vector away from you. Momentum p is mv, angular momentum of a particle is r x p. For an extended body, it turns out that angular momentum is the inertia times the spin rate in radians/second. Click spin directions

for some surprising examples.

Torque exerted by a force on an object is the cross product r x F where r is the distance vector from a reference point (usually taken to be an axis of rotation) to the point where F is applied to the object. This is a measure of the ability of F to produce a rotation about the reference point. The angular acceleration is proportional to the vector sum of all torques. We define inertia (or moment of inertia or rotational inertia) as the constant of proportionality, so the vector sum of all torques acting on a body is equal to inertia times angular acceleration.

S t = Ia

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