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Cross product defined as a determinant

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I am currently reviewing some algebra and I recalled that cross product can be defined in terms of a determinant: wtf i,j and k are described as being an orthonormal basis. We can take i, j and k as the following: i = {1,0,0} j = {0,1,0} k = {0,0,1} However each of these are 3-d vectors, how can they "fit" into their respective matrix positions while preserving the shape of the matrix?

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The operation requires multiplication of vectors by scalars, and addition of vectors, which are both valid operations. That form is just a short hand to help remember the cross product formula. "Fitting" has nothing to do with it.

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It's not a real determinant, just an interesting shorthand / memory aid / coincidence. Just pretend that i, j, and k are normal scalar variables.

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