Can you help me find a unit vector that is orthogonal to both [1, 1, 0] and [1, 0, 1]?

1 Answer
Feb 6, 2018

#[sqrt(3)/3, -sqrt(3)/3, -sqrt(3)/3]#

Explanation:

The cross product of two vectors is orthogonal to both:

#[1, 1, 0] xx [1, 0, 1] = abs((hat(i), hat(j), hat(k)), (1, 1, 0), (1, 0, 1)) = [1, -1, -1]#

Note that the dot product of this vector with either of the two original vectors is #0#.

Now to make it of unit length, we need to divide it by its norm:

#abs("["1, -1, -1"]") = sqrt(1^2+(-1)^2+(-1)^2) = sqrt(3)#

So a suitable unit vector is:

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