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

1 Answer
Narad T.
Nov 10, 2016

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, vecvis 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〉

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