Question

What is the cross product of `[1,-2,-1]` and `[-2,0,3]`?

Answer 1

The answer is `=〈-6,-1,-4〉`

Explanation:

The cross product of 2 vectors, `〈a,b,c〉` and `d,e,f〉`

is given by the determinant

`| (hati,hatj,hatk), (a,b,c), (d,e,f) |`

`= hati| (b,c), (e,f) | - hatj| (a,c), (d,f) |+hatk | (a,b), (d,e) |`

and `| (a,b), (c,d) |=ad-bc`

Here, the 2 vectors are `〈1,-2,-1〉` and `〈-2,0,3〉`

And the cross product is

`| (hati,hatj,hatk), (1,-2,-1), (-2,0,3) |`

`=hati| (-2,-1), (0,3) | - hatj| (1,-1), (-2,3) |+hatk | (1,-2), (-2,0) |`

`=hati(-6+0)-hati(3-2)+hatk(0-4)`

`=〈-6,-1,-4〉`

Verification, by doing the dot product

`〈-6,-1,-4〉.〈1,-2,-1〉=-6+2+4=0`

`〈-6,-1,-4〉.〈-2,0,3〉=12+0-12=0`

Therefore, the vector is perpendicular to the other 2 vectors