a×b×c

a×b×c is a notation that can have different meanings depending on context. In basic arithmetic, if a, b, and c are scalars, a×b×c simply denotes the ordinary product of three numbers, which is commutative and associative: a×b×c = a×(b×c) = (a×b)×c.

In vector algebra, × commonly denotes the cross product, which is defined between two vectors in three-dimensional space. Since the cross product is binary, a×b×c is not a standard operation and is typically interpreted in one of two ways: (a×b)×c or a×(b×c). These two results are generally not equal because the cross product is not associative. There are useful vector identities for these expressions: (a×b)×c = b(a·c) − a(b·c) and a×(b×c) = b(a·c) − c(a·b).

A related concept is the scalar triple product a·(b×c), which yields a scalar equal to the oriented volume of the parallelepiped spanned by a, b, and c. Its magnitude is |a·(b×c)| = |a||b||c| times the sine of the angles in the configuration, with the sign determined by orientation via the right-hand rule.

Common applications of the cross product include computing torques (r×F) and angular momentum (r×p) in physics, and determining normal vectors to surfaces in geometry and computer graphics. In summary, a×b×c may denote a scalar triple product, a scalar product of three factors, or the result of two successive cross products, depending on the context.