What is the cross product of `[-1,0,1]` and `[3, 1, -1]`?

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.

`[-1,0,1] xx [3,1,-1]`

`= (-hati + hatk) xx (3hati + hatj - hatk)`

`= color(white)( (color(black){-hati xx 3hati - hati xx hatj - hati xx (-hatk)}), (color(black){+hatk xx 3hati + hatk xx hatj + hatk xx (-hatk)}) )`

`= color(white)( (color(black){-3(vec0) - hatk - hatj}), (color(black){+ 3hatj qquad - hati - vec0}) )`

`= -hati + 2hatj + -1hatk`

`= [-1,2,-1]`