Question

What is the cross product of `<7, 5 ,6 >` and `<3 ,5 ,-6 >`?

Answer 1

The answer is `=〈-60,60,20〉`

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 `〈7,5,6〉` and `〈3,5,-6〉`

And the cross product is

`| (hati,hatj,hatk), (7,5,6), (3,5,-6) |`

`=hati| (5,6), (5,-6) | - hatj| (7,6), (3,-6) |+hatk | (7,5), (3,5) |`

`=hati(-30-30)-hati(-42-18)+hatk(35-15)`

`=〈-60,60,20〉`

Verification, by doing the dot product

`〈-60,60,20〉.〈7,5,6〉=-420+300+120=0`

`〈-60,60,20〉.〈3,5,-6〉=-180+300-120=0`

Therefore, the vector is perpendicular to the other 2 vectors