Vector/cross product, once and for all

It is neither surprising nor satisfactory to see that there is a fair amount of struggle about the vector product, i.e. the cross product. It remains absolutely true that the first two pages of Lec_11-10.pdf contain all there is to know about the vector product at this point of your life. However, I like to go over the vector product one more time to help those of you who are struggling with this.

Given two vectors A and B how do we obtain the vector/cross prodcut, A x B?
(1) Geometric method
(i) Align the tails of the two vectors, by parallel-shifting one of them.
(ii) Imagine rotating A to B with the tail fixed.
(iii) The right-hand rule is then applied to figure out the direction of A x B.
(iv) The magnitude is given by ABsin(theta), where theta is the angle between A and B.
(2) Numerical mothod
A = (Ax, Ay, Az), B = (Bx, By, Bz) => A x B = (AyBz - AzBy, AzBx - AxBz, AxBy - AyBx)

These two methods are equivalent to each other, of course. In my course, you are not asked to use (2). In later courses of Phys 6, you might have a chance to do that.

Here is a simple example, in the context of the torque.