a×c

In mathematics, the expression a×c can denote the product of two entities named a and c. The exact interpretation depends on the context and the nature of a and c. Most commonly, a×c denotes either scalar multiplication of numbers or a cross product of vectors in three dimensions.

If a and c are real or complex numbers, a×c is the ordinary product. It is commutative (a×c = c×a), associative, and distributive over addition (a×(b+c) = a×b + a×c). The result is a scalar. The notation with × is conventional though some texts use either the dot product or juxtaposition for multiplication.

If a and c are three-dimensional vectors, a×c denotes the cross product. The result is a vector perpendicular to both, with magnitude equal to |a||c| sin θ, where θ is the angle between a and c. The direction is given by the right-hand rule and the cross product is anti-commutative: a×c = −(c×a). When a and c are parallel, a×c = 0. In coordinates, if a=(a1,a2,a3) and c=(c1,c2,c3), a×c = (a2 c3 − a3 c2, a3 c1 − a1 c3, a1 c2 − a2 c1).

In physics and engineering, the cross product appears in torque, angular momentum, and rotational dynamics. The symbol can also denote other products in advanced algebra, depending on definitions used in a given field; thus disambiguation by context is essential.