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

1 Answer
Nov 10, 2016

Answer:

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〉#