the cross product

The dot product takes two vectors and returns a number. The cross product takes two vectors in space and returns a brand-new VECTOR — one that points straight out of the plane the two vectors lie in. It is the natural tool for 'perpendicular to both at once' and for measuring the area they sweep out. Torque, angular momentum, and the normal to a plane all live here.

Given u and v in three dimensions, their cross product u x v is the vector with two defining properties. First, its direction is perpendicular to BOTH u and v, with the side chosen by the right-hand rule: point the fingers of your right hand along u and curl them toward v, and your thumb points along u x v. Second, its magnitude is |u x v| = |u| |v| sin theta, which equals the area of the parallelogram spanned by u and v. In components there is a tidy formula: for u = (u_1, u_2, u_3) and v = (v_1, v_2, v_3), u x v = (u_2 v_3 - u_3 v_2, u_3 v_1 - u_1 v_3, u_1 v_2 - u_2 v_1). The magnitude vanishes exactly when sin theta = 0, that is when u and v are parallel (or one is zero) — parallel arrows span no area.

Because u x v is automatically perpendicular to both inputs, it is the standard way to manufacture a normal vector for a plane through three points, or to find a direction perpendicular to two given lines. The half-magnitude (1/2)|u x v| is the area of the triangle with sides u and v. One sharp warning: the cross product is NOT commutative — swapping the inputs reverses the result, u x v = -(v x u), because the right-hand rule flips. It is also not associative. And it is special to three dimensions (and a quirky case in seven); there is no ordinary cross product of two vectors in the plane or in higher dimensions. Treat the order and the dimension with respect.