, where φ is the angle between B and C in their plane,

**n**_{(BxC)} is the unit vector perpendicular to their plane, ω is the angle between A and the unit vector

**n**_{(BxC)} in their plane, and

**n**_{(Ax(BxC)} is the unit vector perpendicular to

**A** and the unit vector

**n**_{(BxC)} in their plane.

This can only be zero when at least one of φ, ω, A, B, or C = 0.

Since there are cases where none of φ, ω, A, B, or C is equal to 0, the statement (

**A** x

**B**) x

**C** =

**A** x (

**B** x

**C**) is not true for all vectors

**A**,

**B** and

**C** and therefore we can conclude that the cross product is not associative.

Is this a valid proof? Would it be considered an attempt at a direct proof? I used a particular example, but I didn't use a contradiction, so I'm thinking yes, but I wouldn't know.