Question

What is the cross product of `[4, -4, 4]` and `[-6, 5, 1]`?

Answer 1

\begin{pmatrix}-24&-28&-4\end{pmatrix}

Explanation:

Use the following cross product formula:

`(u1,u2,u3)xx(v1,v2,v3) = (u2v3 − u3v2 , u3v1 − u1v3 , u1v2 − u2v1)`

`(4,-4,4)xx(-6,5,1) = (-4*1 − 4*5 , 4*-6 − 4*1 , 4*5 − -4*-6)`

`=(-24,-28,-4)`

Answer 2

The vector is `= 〈-24,-28,-4〉`

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,-4,4〉` and `vecb=〈-6,5,1〉`

Therefore,

`| (veci,vecj,veck), (4,-4,4), (-6,5,1) |`

`=veci| (-4,4), (5,1) | -vecj| (4,4), (-6,1) | +veck| (4,-4), (-6,5) |`

`=veci((-4)*(1)-(5)*(4))-vecj((4)*(1)-(-6)*(4))+veck((4)*(5)-(-4)*(-6))`

`=〈-24,-28,-4〉=vecc`

Verification by doing 2 dot products

`〈4,-4,4〉.〈-24,-28,-4〉=(4)*(-24)+(-4)*(-28)+(4)*(-4)=0`

`〈-24,-28,-4〉.〈-6,5,1〉=(-24)*(-6)+(-28)*(5)+(-4)*(1)=0`

So,

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