Question

What is the cross product of `[-1, -1, 2]` and `[2, 5, 4]`?

Answer 1

`[-14,8,-3]`

Explanation:

We know that `vecA xx vecB = ||vecA|| * ||vecB|| * sin(theta) hatn`, where `hatn` is a unit vector given by the right hand rule.

So for of the unit vectors `hati`, `hatj` and `hatk` in the direction of `x`, `y` and `z` respectively, we can arrive at the following results.

`color(white)( (color(black){hati xx hati = vec0}, color(black){qquad hati xx hatj = hatk}, color(black){qquad hati xx hatk = -hatj}), (color(black){hatj xx hati = -hatk}, color(black){qquad hatj xx hatj = vec0}, color(black){qquad hatj xx hatk = hati}), (color(black){hatk xx hati = hatj}, color(black){qquad hatk xx hatj = -hati}, color(black){qquad hatk xx hatk = vec0}))`

Another thing that you should know is that cross product is distributive, which means

`vecA xx (vecB + vecC) = vecA xx vecB + vecA xx vecC`.

We are going to need all of these results for this question.

`[-1,-1,2] xx [2,5,4]`

`= (-hati - hatj + 2hatk) xx (2hati + 5hatj + 4hatk)`

`= color(white)( (color(black){-hati xx 2hati - hati xx 5hatj - hati xx 4hatk}), (color(black){-hatj xx 2hati - hatj xx 5hatj - hatj xx 4hatk}), (color(black){+2hatk xx 2hati + 2hatk xx 5hatj + 2hatk xx 4hatk}) )`

`= color(white)( (color(black){-2(vec0) - 5hatk + 4hatj}), (color(black){+2hatk quad - 5(vec0) - 4hati}), (color(black){quad +4hatj quad - 10hati + 8(vec0)}) )`

`= -14hati + 8hatj + 8hatk`

`= [-14,8,-3]`