Is the vector cross product only defined for 3D?

Wikipedia introduces the vector product for two vectors a and b as
It then mentions that \vec n is the vector normal to the plane made by \vec a and \vec b, implying that \vec a and \vec b are 3D vectors. Wikipedia mentions something about a 7D cross product, but I’m not going to pretend I understand that.

My idea, which remains unconfirmed with any source, is that a cross product can be thought of a vector which is orthogonal to all vectors which you are crossing. If, and that’s a big IF, this is right over all dimensions, we know that for a set of n-1 n-dimensional vectors, there exists a vector which is orthogonal to all of them. The magnitude would have something to do with the area/volume/hypervolume/etc. made by the vectors we are crossing.

Am I right to guess that this multidimensional aspect of cross vectors exists or is that last part utter rubbish?


Yes, you are correct. You can generalize the cross product to n dimensions by saying it is an operation which takes in n-1 vectors and produces a vector that is perpendicular to each one. This can be easily defined using the exterior algebra and Hodge star operator the cross product of v_1,\ldots,v_{n-1} is then just *(v_1 \wedge v_2 \cdots \wedge v_{n-1}).

Then the magnitude of the cross product of n-1 vectors is the volume of the higher-dimensional parallelogram that they determine. Specifying the magnitude and being orthogonal to each of the vectors narrows the possiblity to two choices– an orientation picks out one of these.

Source : Link , Question Author : VF1 , Answer Author : Eric O. Korman

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