Question

What is the cross product of `<4 , 5 ,-7 >` and `<5 ,1 ,-3 >`?

Answer 1

The vector is `=〈-8,-23,-21〉`

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,-7〉` and `vecb=〈5,1,-3〉`

Therefore,

`| (veci,vecj,veck), (4,5,-7), (5,1,-3) |`

`=veci| (5,-7), (1,-3) | -vecj| (4,-7), (5,-3) | +veck| (4,5), (5,1) |`

`=veci((5)*(-3)-(-7)*(1))-vecj((4)*(-3)-(-7)*(5))+veck((4)*(1)-(5)*(5))`

`=〈-8,-23,-21〉=vecc`

Verification by doing 2 dot products

`〈-8,-23,-21〉.〈4,5,-7〉=(-8)*(4)+(-23)*(5)+(-21)*(-7)=0`

`〈-8,-23,-21〉.〈5,1,-3〉=(-8)*(5)+(-23)*(1)+(-21)*(-3)=0`

So,

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