How do you find a unit vector that is orthogonal to both u = (1, 0, 1) v = (0, 1, 1)?

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Answer 1 · 2016-07-23T07:15:20

#(u xx v) / (|| u xx v ||) = (-sqrt(3)/3, -sqrt(3)/3, sqrt(3)/3)#

The cross product of #u = (u_1, u_2, u_3)# and #v = (v_1, v_2, v_3)# is given by:

#(u_1, u_2, u_3) xx (v_1, v_2, v_3) = (abs((u_2, u_3),(v_2, v_3)), abs((u_3, u_1),(v_3, v_1)), abs((u_1, u_2),(v_1, v_2)))#

This will be orthogonal to both #u# and #v#, but will need scaling to make it unit length.

So we find:

#u xx v = (1, 0, 1) xx (0, 1, 1)#

#= (abs((0, 1),(1, 1)), abs((1, 1),(1, 0)), abs((1, 0),(0,1)))#

#= (-1, -1, 1)#

Then:
#||""(-1, -1, 1) || = sqrt((-1)^2+(-1)^2+1^2) = sqrt(1+1+1) = sqrt(3)#

So to make #(-1, -1, 1)# into a unit vector, divide it by #sqrt(3)#:

#1/sqrt(3) (-1, -1, 1) = sqrt(3)/3 (-1, -1, 1) = (-sqrt(3)/3, -sqrt(3)/3, sqrt(3)/3)#