information:classification
Cross product

The cross product (also called the vector product or outer product) is only meaningful in three or seven dimensions. The cross product differs from the dot product primarily in that the result of the cross product of two vectors is a vector. The cross product, denoted a × b, is a vector perpendicular to both a and b and is defined as

a × b = ‖ a ‖ ‖ b ‖ sin (θ) n {\displaystyle \mathbf {a} \times \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\sin(\theta)\,\mathbf {n} }

where θ is the measure of the angle between a and b, and n is a unit vector perpendicular to both a and b which completes a right-handed system. The right-handedness constraint is necessary because there exist two unit vectors that are perpendicular to both a and b, namely, n and (− n).

The cross product a × b is defined so that a, b, and a × b also becomes a right-handed system (although a and b are not necessarily orthogonal). This is the right-hand rule.

The length of a × b can be interpreted as the area of the parallelogram having a and b as sides.

The cross product can be written as a × b = (a 2 b 3 − a 3 b 2) e 1 + (a 3 b 1 − a 1 b 3) e 2 + (a 1 b 2 − a 2 b 1) e 3. {\displaystyle {\mathbf {a} }\times {\mathbf {b} }=(a_{2}b_{3}-a_{3}b_{2}){\mathbf {e} }_{1}+(a_{3}b_{1}-a_{1}b_{3}){\mathbf {e} }_{2}+(a_{1}b_{2}-a_{2}b_{1}){\mathbf {e} }_{3}.}

For arbitrary choices of spatial orientation (that is, allowing for left-handed as well as right-handed coordinate systems) the cross product of two vectors is a pseudovector instead of a vector (see below).