Question

What is the cross product of `[0,8,5]` and `[1,2,-4]`?

Answer 1

`[0,8,5] xx [1,2,-4] = [-42,5,-8]`

Explanation:

The cross product of `vecA` and `vecB` is given by

`vecA xx vecB = ||vecA|| * ||vecB|| * sin(theta) hatn`,

where `theta` is the positive angle between `vecA` and `vecB`, and `hatn` is a unit vector with direction given by the right hand rule.

For the unit vectors `hati`, `hatj` and `hatk` in the directions of `x`, `y` and `z` respectively,

`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}))`

Also, cross product is distributive, which means

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

For this question,

`[0,8,5] xx [1,2,-4]`

`= (8hatj + 5hatk) xx (hati + 2hatj - 4hatk)`

`= color(white)( (color(black){qquad 8hatj xx hati + 8hatj xx 2hatj + 8hatj xx (-4hatk)}), (color(black){+5hatk xx hati + 5hatk xx 2hatj + 5hatk xx (-4hatk)}) )`

`= color(white)( (color(black){-8hatk + 16(vec0) - 32hati}), (color(black){qquad +5hatj - quad 10hati quad - 20(vec0)}) )`

`= -42hati + 5hatj - 8hatk`

`= [-42,5,-8]`