1. The problem statement, all variables and given/known data Show that any vector V in a plane can be written as a linear combination of two non-parallel vectors A and B in the plane; that is, find a and b so that V = aA + bB. Take components perpendicular to the plane to show that a = (B x V)[itex]\bullet[/itex]n / (B x A)[itex]\bullet[/itex]n 2. Relevant equations 3. The attempt at a solution (Upper case letters are vectors, it gets tiring to bold everything) BxV = B x (aA + bB) = a(BxA) + b(BxB) = a(BxA) BxV = a(BxA) Now, if I compare this to the solution given for a, I see that they took the dot product with n on both side, so (BxV)[itex]\bullet[/itex]n = a(BxA)[itex]\bullet[/itex]n It is then trivial to isolate a and find the right answer. However, I was wondering if I could take the dot product with any vector, since technically I'd be applying the same operation of both sides of the equation. If not, why does this only work with the unit vector perpendicular to the plane?