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Vector cross products in n dimensions

All the usual properties of the vector cross product a1 x a2on E 3 (three‐dimensional Euclidean space) are derived from an algebraic definition of a1 x a2. The obvious generalization of this definition yields a vector cross product a1 x ... x a n‐1 on En, and we show that all the usual properties of a1 x a2 receive natural generalizations. Even the familiar results in E 3 involving the repeated vector product (a x b) x c, and solution of the equation a x v = b, generalize in an exceedingly straightforward fashion. Finally we enquire whether or not other n‐dimensional generalizations of the three‐dimensional vector cross product are possible, pointing out exceptional features of the cases n = 7 and n = 8.