Question

What is the cross product of `[3, 0, 5]` and `[2, -1, 1]`?

Answer 1

The vector is `=〈5,7,-3〉`

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=〈3,0,5〉` and `vecb=〈2,-1,1〉`

Therefore,

`| (veci,vecj,veck), (3,0,5), (2,-1,1) |`

`=veci| (0,5), (-1,1) | -vecj| (3,5), (2,1) | +veck| (3,0), (2,-1) |`

`=veci((0)*(1)-(-1)*(5))-vecj((3)*(1)-(2)*(5))+veck((3)*(-1)-(0)*(2))`

`=〈5,7,-3〉=vecc`

Verification by doing 2 dot products

`〈5,7,-3〉.〈3,0,5〉=(5)*(3)+(7)*(0)+(-3)*(5)=0`

`〈5,7,-3〉.〈2,-1,1〉=(5)*(2)+(7)*(-1)+(-3)*(1)=0`

So,

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