Question

What is the cross product of `<5, -3, 8 >` and `<-2 ,7 ,3 >`?

Answer 1

`-65hati - 31hatj + 29hatk`

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.

`<5,-3,8> xx <-2,7,3>`

`= (5hati - 3hatj + 8hatk) xx (-2hati + 7hatj + 3hatk)`

`= color(white)( (color(black){qquad 5hati xx (-2hati) + 5hati xx 7hatj + 5hati xx 3hatk}), (color(black){-3hatj xx (-2hati) - 3hatj xx 7hatj - 3hatj xx 3hatk}), (color(black){+8hatk xx (-2hati) + 8hatk xx 7hatj + 8hatk xx 3hatk}) )`

`= color(white)( (color(black){-10(vec0) + 35hatk qquad - 15hatj}), (color(black){-6hatk qquad - 21(vec0) - 9hati}), (color(black){qquad -16hatj qquad - 56hati qquad + 24(vec0)}) )`

`= -65hati - 31hatj + 29hatk`