Question

What is the cross product of `<0,8,5>` and `<-1,-1,2>`?

Answer 1

`<21,-5,8>`

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.

`<0,8,5> xx <-1,-1,2>`

`= (8hatj + 5hatk) xx (-hati - hatj + 2hatk)`

`= color(white)( (color(black){qquad 8hatj xx (-hati) + 8hatj xx (-hatj) + 8hatj xx 2hatk}), (color(black){+5hatk xx (-hati) + 5hatk xx (-hatj) + 5hatk xx 2hatk}) )`

`= color(white)( (color(black){8hatk - 8(vec0) + 16hati}), (color(black){-5hatj + 5hati qquad + 10(vec0)}) )`

`= 21hati - 5hatj + 8hatk`

`= <21,-5,8>`