The vector perpendicular to the plane containing 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,-3,-1〉 and vecb=〈1,4,-2〉
Therefore,
| (veci,vecj,veck), (2,-3,-1), (1,4,-2) |
=veci| (-3,-1), (4,-2) | -vecj| (2,-1), (1,-2) | +veck| (2,-3), (1,4) |
=veci(-3*-2+1*4)-vecj(-2*2+1*1)+veck(2*4+3*1)
=〈10,3,11〉=vecc
Verification by doing 2 dot products
〈10,3,11〉.〈2,-3,-1〉=10*2-3*3-11*1=0
〈10,3,11〉.〈1,4,-2〉=10*1+3*4-11*2=0
So,
vecc is perpendicular to veca and vecb
The unit vector is
=vecc/(||vecc||)
=(〈10,3,11〉)/(||〈10,3,11〉||)
=1/sqrt(100+9+121)〈10,3,11〉
=1/sqrt230〈10,3,11〉
