Can someone please explain to me the motivation for these definitions of the cross product? Let A = (a1, a2, a3), B = (b1, b2, b3). Let A and B be the magnitudes of A and B, respectively, and let [itex]\theta[/itex] be the angle between the vectors. A X B = AB sin([itex]\theta[/itex]) A X B = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1) As I understand it, the cross product was developed for application to situations in physics, in which two vectors produce a result perpendicular to both of them. Therefore, the cross product would be orthogonal to both vectors: A [itex]\cdot[/itex] (A X B) = 0, and B [itex]\cdot[/itex] (A X B) = 0. However, I do not understand the other properties of the cross product which led to these definitions. Thank you for your help.