Question

What is the cross product of `<-3,5,8 >` and `<6, -2, 7 >`?

Answer 1

The vector is `=〈51,69,-24〉=`

Explanation:

The cross product of 2 vectors is calculated with the determinant

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

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

Here, we have `veca=〈-3,5,8〉` and `vecb=〈6,-2,7〉`

Therefore,

`| (veci,vecj,veck), (-3,5,8), (6,-2,7) |`

`=veci| (5,8), (-2,7) | -vecj| (-3,8), (6,7) | +veck| (-3,5), (6,-2) |`

`=veci(5*7+2*8)-vecj(-3*7-6*8)+veck(3*2-5*6)`

`=〈51,69,-24〉=vecc`

Verification by doing 2 dot products

`〈51,69,-24〉.〈-3,5,8〉=-51*3+69*5-24*8=0`

`〈51,69,-24〉.〈6,-2,7〉=51*6-69*2-24*7=0`

So,

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

Answer 2

`=(51,69,-24).`

Explanation:

Let, `vecx=(-3,5,8)=-3i+5j+8k,` and,

`vecy=(6,-2,7)=6i-2j+7k.`

`:. vecx xx vecy=|(i,j,k),(-3,5,8),(6,-2,7)|`

`=(35-(-16))i-(-21-48)j+(6-30)k`

`=51i+69j-24k`

`=(51,69,-24).`