Question

What is the cross-product of two vectors?

Answer 1

A vector orthogonal to both of them...

Explanation:

The cross product of two vectors in `3` dimensional space is a third vector orthogonal to both of them and of length proportional to the product of the lengths of the two vectors.

We write the cross product of `vec(u) = < u_1, u_2, u_3 >` and `vec(v) = < v_1, v_2, v_3 >` as:

`vec(u) xx vec(v) = < u_2v_3-u_3v_2, color(white)(.)u_3v_1-u_1v_2, color(white)(.)u_1v_2-u_2v_1 >`

If the angle between the vectors `vec(u)` and `vec(v)` is `theta` then we find:

`abs(abs(vec(u) xx vec(v))) = abs(abs(vec(u))) * abs(abs(vec(v))) color(white)(.)sin theta`

Another way of writing the cross product is:

`(u_1hat(i) + u_2hat(j) + u_3hat(k)) xx (v_1hat(i)+v_2hat(j)+v_3hat(k)) = abs((hat(i), hat(j), hat(k)), (u_1, u_2, u_3), (v_1, v_2, v_3))`

Note that if `vec(u)` and `vec(v)` are parallel, then their cross product is the zero vector.

See also https://socratic.org/s/aPeBh7vZ