Question

Determine unit vector which is perpendicular to both A=2i+j+k and B=i-j+2k?

Answer 1

We know that cross product of any two vectors yields a vector which is perpendicular to both vectors
`:.` for two vectors `vecA and vecB` if `vecC` is the vector perpendicular to both.
`vecC=vecAxxvecB=` `[(hati, hatj, hatk), (A_1, A_2,A_3),(B_1, B_2, B_3)]`
`=(A_2B_3−B_2A_3)hati−(A_1B_3−B_1A_3)hatj+(A_1B_2−B_1A_2)hatk`.
Inserting given vectors we obtain
`vecC=` `[(hati, hatj, hatk), (2, 1,1),(1, -1, 2)]`
`=(1xx2−(-1)xx1)hati−(2xx2−1xx1)hatj+(2xx(-1)−1xx1)hatk`.
`=3hati−3hatj−3hatk`.

Now unit vector in the direction of `vecC` is `vecC/|vecC|`
`:.|vecC|=sqrt(3^2+(-3)^2+(-3)^2)`
`=sqrt27`
`=3sqrt3`
Therefore desired unit vector is
`1/sqrt3(hati−hatj−hatk)`