Question

What is the cross product of `(14i - 7j - 7k)` and `(-5i + 12j + 2 k)`?

Answer 1

`70hati + 7hatj + 133hatk`

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.

`(14hati - 7hatj - 7hatk) xx (-5hati + 12hatj + 2hatk)`

`= color(white)( (color(black){qquad 14hati xx (-5hati) + 14hati xx 12hatj + 14hati xx 2hatk}), (color(black){-7hatj xx (-5hati) - 7hatj xx 12hatj - 7hatj xx 2hatk}), (color(black){-7hatk xx (-5hati) - 7hatk xx 12hatj - 7hatk xx 2hatk}) )`

`= color(white)( (color(black){-70(vec0) + 168hatk qquad - 28hatj}), (color(black){-35hatk qquad - 84(vec0) - 14hati}), (color(black){qquad +35hatj qquad + 84hati qquad - 14(vec0)}) )`

`= 70hati + 7hatj + 133hatk`