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

1 Answer
Sep 15, 2016

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)#