Question

What is the cross product of `[-1, -1, 2]` and `[-1, 2, 2]`?

Answer 1

`[-1,-1,2] xx [-1,2,2] = [-6, 0, -3]`

Explanation:

The cross product between two vectors `vecA` and `vecB` is defined to be

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

where `hatn` is a unit vector given by the right hand rule, and `theta` is the angle between `vecA` and `vecB` and must satisfy `0<=theta<=pi`.

For of the unit vectors `hati`, `hatj` and `hatk` in the direction of `x`, `y` and `z` respectively, using the above definition of cross product gives the following set of 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}))`

Also, note that cross product is distributive.

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

So for this question.

`[-1,-1,2] xx [-1,2,2]`

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

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

`= color(white)( (color(black){vec0 - 2hatk quad qquad + 2hatj}), (color(black){-hatk - 2(vec0) - 2hati}), (color(black){- 2hatj - 4hati quad - 4(vec0)}) )`

`= -6hati - 3hatk`

`= [-6,0,-3]`