Question

What is the unit vector that is orthogonal to the plane containing `<3, 2, 1>` and `<1, 1, 1>`?

Answer 1

The unit vector is `=1/sqrt6〈1,-2,1〉`

Explanation:

To calculate a vector ortogonal to 2 other vectors, we caculate the cross product.
The cross product is the determinant of `∣((veci,vecj,veck),(3,2,1),(1,1,1))∣`

`=veci(2-1)-vecj(3-1)+veck(3-2)`
So the vector ortogonal is `vecv=〈1,-2,1〉`
To verify, we do the dot products .
`〈3,2,1〉.〈1,-2,1〉=3-4+1=0`
`〈1,1,1〉.〈1,-2,1〉=1-2+1=0`
As the dot products are `=0`, `vecv`is ortogonal to the other 2 vectors.
To compute the unit vector, we divide by the modulus.

`hatv=vecv/(∣vecv∣)`
The modulus is `=sqrt(1+4+1)=sqrt6`

Therefore, `hatv=1/sqrt6〈1,-2,1〉`