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## Main Question or Discussion Point

Did someone just realize that taking the determinant of a specific matrix gives you the cross product formula, or is there is a specific conceptual reason why it works?

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Did someone just realize that taking the determinant of a specific matrix gives you the cross product formula, or is there is a specific conceptual reason why it works?

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It can be shown that for a finite dimensional real vector space with an inner product (like the dot product in R^3) any real linear function is of the form f(v)=u.v for some vector u. So if we consider the determinant of a 3x3 matrix as a function of its first column then this must be a dot product with some vector which depends on the other two columns. (By direct calculation it is the dot product with the vector whose components are 2x2 determinants, but you seemed to be asking for a more conceptual explanation). Since the determinant is linear in each column and antisymmetric (changes sign when any two rows or colums are exchanged) the vector whose dot product with the first column gives the determinant depends linearly on each of the other two columns and changes sign when they are reversed. Also since the determinant is zero when columns repeat, this combination of the last two columns is orthogonal to both of them (since the corresponding function of the first column is zero when it matches either of the other two columns). These conditions are enough to determine the cross product.

[More generally, in an n-dimensional space there is a generalization of the cross product defined not for pairs but for groups of n-1 vectors by taking as components the signed minor determinants which multiply by the entries of the first column to get the overall determinant of the nxn matrix with the given n-1 vectors as its remaining columns]

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tiny-tim

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Hi

I thinkDid someone just realize that taking the determinant of a specific matrix gives you the cross product formula, or is there is a specific conceptual reason why it works?

a determinant is usually a number, and is derived from a transformation matrix …

so

I can't think of a good answer to any of this, I can only think of more problems

… for example, the determinant of a product of two transformations is the product of the determinants, but what would the first "product" mean in this case?

… for example, the determinant of a product of two transformations is the product of the determinants, but what would the first "product" mean in this case?

Does anybody have a good answer to this?

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