, 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 C
) is not true for all vectors A
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.