Cross product in three-dimensional Euclidean space

In the standard oriented coordinates of Euclidean space $\mathbb R^3$ , the cross product is

a\times b=(a_2b_3-a_3b_2,\ a_3b_1-a_1b_3,\ a_1b_2-a_2b_1).

It is bilinear, satisfies $a\times b=-b\times a$ , and obeys $(a\times b)\cdot c=\det[a\ b\ c]$ . Thus it is perpendicular to both inputs. Reversing the orientation reverses the sign of the product.

Vector identity

Direct expansion gives $a\times(b\times c)=b(a\cdot c)-c(a\cdot b)$ . In particular, $a\times(a\times b)=-|a|^2b$ when $a\cdot b=0$ . This identity recovers a transverse vector from a curl symbol in Fourier calculations.