Question

What is the cross product of `<< -1, -1, 2 >>` and `<< 4,3,6 >>`?

Answer 1

Well, you have at least two ways to do it.

The first way:

Let `vecu = << u_1,u_2,u_3 >>` and `vecv = << v_1,v_2,v_3 >>`. Then:

`color(blue)(vecu xx vecv) = << u_2v_3 - u_3v_2, u_3v_1 - u_1v_3, u_1v_2 - u_2v_1 >>`

`= << -1*6 - 2*3, 2*4 - (-1*6), -1*3 - (-1*4) >>`

`= color(blue)(<< -12, 14, 1 >>)`

Assuming you didn't know that formula, the second way (which is a little more foolproof) is recognizing that:

`hati xx hatj = hatk`
`hatj xx hatk = hati`
`hatk xx hati = hatj`
`hatA xx hatA = vec0`
`hatA xx hatB = -hatB xx hatA`

where `hati = << 1,0,0 >>`, `hatj = << 0,1,0 >>`, and `hatk = << 0,0,1 >>`.

Thus, rewriting the vectors in unit vector form:

`(-hati - hatj + 2hatk)xx(4hati + 3hatj + 6hatk)`

`= cancel(-4(hati xx hati))^(0) - 3(hati xx hatj) - 6(hati xx hatk) - 4(hatj xx hati) - cancel(3(hatj xx hatj))^(0) - 6(hatj xx hatk) + 8(hatk xx hati) + 6(hatk xx hatj) + cancel(12(hatk xx hatk))^(0)`

`= -3hatk + 6hatj + 4hatk - 6hati + 8hatj - 6hati`

`= - 12hati + 14hatj + hatk`

`= color(blue)(<< -12, 14, 1 >>)`

as expected.