Question

What is the cross product of `<4 , 5 ,-9 >` and `<4, 3 ,-3 >`?

Answer 1

The vector is `<12, -24, -8>`

Explanation:

The cross product of 2 vectors is calculated with the determinant

`| (veci,vecj,veck), (d,e,f), (g,h,i) |`

where `veca=〈d,e,f〉` and `vecb=〈g,h,i〉` are the 2 vectors

Here, we have `veca=〈4,5,-9〉` and `vecb=〈4,3,-3〉`

Therefore,

`| (veci,vecj,veck), (4,5,-9), (4,3,-3) |`

`=veci| (5,-9), (3,-3) | -vecj| (4,-9), (4,-3) | +veck| (4,5), (4,3) |`

`=veci((5)*(-3)-(9)*(-3))-vecj((4)*(-3)-(9)*(4))+veck((4)*(3)-(4)*(5))`

`=〈12,-24,-8〉=vecc`

Verification by doing 2 dot products

`〈12,-24,-8〉.〈4,5,-9〉=(12)*(4)+(-24)*(5)+(-8)*(-9)=0`

`〈12,-24,-8〉.〈4,3,-3〉=(12)*(4)+(-24)*(3)+(-8)*(-3)=0`

So,

`vecc` is perpendicular to `veca` and `vecb`