Question

What is the cross product of `<-1, 2 ,27 >` and `<-3 ,1 ,-1 >`?

Answer 1

The answer is `=〈-29,-82,5〉`

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 `〈-1,2,27〉` and `〈-3,1,-1〉`

And the cross product is

`| (hati,hatj,hatk), (-1,2,27), (-3,1,-1) |`

`=hati| (2,27), (1,-1) | - hatj| (-1,27), (-3,-1) |+hatk | (-1,2), (-3,1) |`

`=hati(-2-27)-hati(1+81)+hatk(-1+6)`

`=〈-29,-82,5〉`

Verification, by doing the dot product

`〈-29,-82,5〉.〈-1,2,27〉=29-164+135=0`

`〈-29,-82,5〉.〈-3,1,-1〉=87-82-5=0`

Therefore, the vector is perpendicular to the other 2 vectors