Question

What is the cross product of `[2, 5, 4]` and `[1, -4, 0]`?

Answer 1

`[16,4,-13].`

Explanation:

`[2,5,4]xx[1,-4,0]=|(i,j,k),(2,5,4),(1,-4,0)|,`

`=16i+4j-13k,`

`=[16,4,-13].`

Answer 2

The vector is `=〈16,4,-13〉`

Explanation:

The vector perpendicular to 2 vectors is calculated with the determinant (cross product)

`| (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=〈2,5,4〉` and `vecb=〈1,-4,0〉`

Therefore,

`| (veci,vecj,veck), (2,5,4), (1,-4,0) |`

`=veci| (5,4), (-4,0) | -vecj| (2,4), (1,0) | +veck| (2,5), (1,-4) |`

`=veci(16)-vecj(-4)+veck(-13)`

`=〈16,4,-13〉=vecc`

Verification by doing 2 dot products

`veca.vecc`

`=〈2,5,4>.〈16,4,-13〉=32+20-52=0`

`vecb.vecc`

`=〈1,-4,0〉.〈16,4,-13〉=16-16+0=0`

So,

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