Question

What is the cross product of `<10 ,4 ,1 >` and `<-5 ,2 ,3 >`?

Answer 1

The vector is `=〈10,-35,40〉`

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=〈10,4,1〉` and `vecb=〈-5,2,3〉`

Therefore,

`| (veci,vecj,veck), (10,4,1), (-5,2,3) |`

`=veci| (4,1), (2,3) | -vecj| (10,1), (-5,3) | +veck| (10,4), (-5,2) |`

`=veci((4)*(3)-(1)*(2))-vecj((10)*(3)+(5)*(1))+veck((10)*(2)-(4)*(-5))`

`=〈10,-35,40〉=vecc`

Verification by performing 2 dot products

`〈10,4,1〉.〈10,-35,40〉=(10)*(10)+(4)*(-35)+(1)*(40)=0`

`〈-5,2,3〉.〈10,-35,40〉=(-5)*(10)+(2)*(-35)+(3)*(40)=0`

So,

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