Question

What is the cross product of `<3 ,1 ,-6 >` and `<7 ,3 ,2 >`?

Answer 1

The cross product is `=〈20,-48,2〉`

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 `〈3,1,-6〉` and `〈7,3,2〉`

And the cross product is

`| (hati,hatj,hatk), (3,1,-6), (7,3,2) |`

`=hati| (1,-6), (3,2) | - hatj| (3,-6), (7,2) |+hatk | (3,1), (7,3) |`

`=hati(2+18)-hati(6+42)+hatk(9-7)`

`=〈20,-48,2〉`

Verification, by doing the dot product

`〈20,-48,2〉.〈3,1,-6〉=60-48-12=0`

`〈20,-48,2〉.〈7,3,2〉=140-144+4=0`

Therefore, the vector is perpendicular to the other 2 vectors