Question

How do you find a unit vector that is orthogonal to both u= - 6i + 4j + k and v= 3i + j + 5k?

Answer 1

The unit vector is `=1/1774〈19,33,-18〉`

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=〈-6,4,1〉` and `vecb=〈3,1,5〉`

Therefore,

`| (veci,vecj,veck), (-6,4,1), (3,1,5) |`

`=veci| (4,1), (1,5) | -vecj| (-6,1), (3,5) | +veck| (-6,4), (3,1) |`

`=veci(4*5-1*1)-vecj(-6*5-1*3)+veck(-6*1-4*3)`

`=〈19,33,-18〉=vecc`

Verification by doing 2 dot products

`〈19,33,-18〉.〈-6,4,1〉=-6*19+4*33-1*18=0`

`〈19,33,-18〉.〈3,1,5〉=19*3+1*33-5*18=0`

So,

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

The unit vector is

`hatc=vecc/(||vecc||)=1/sqrt(19^2+33^2+18^2)*〈19,33,-18〉`

`=1/1774〈19,33,-18〉`