If e1 and e2 are vectors in ℝ3 show that e1 x e2 = e3, e2 x e3 = e1 and e3 x e1 = e2. I have tried to prove this but I can't get it. My attempt: Step 1: [a1, a2, a3] x [b1, b2, b3] = [a2b3-a3b2, a3b1-a1b3, a1b2-a2b1] Step 2: [b1, b2, b3] x [a2b3-b2a3, a3b1-a1b3, a1b2-a2b1] = [b2(a1b2-a2b1)-b3(a3b1-a1b3), b3(a2b3-a3b2)-b1(a1b2-a2b1), b1(a3b1-a1b3)-b2(a2b3-b2a3)] ... nothing cancels and I do not end up with [a1, a2, a3], which I should shouldn't I? I also try this with actual numbers but it still doesn't work. Am I doing something completely wrong? I understand, geometrically, why this should happen because the cross product is orthogonal to the two vectors. Am I doing something wrong or is my text book wrong? (I am pretty sure the former is the right answer to that question).