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The cross product $a \times b$ can be represented by the determinant

$$\mathbf{a}\times\mathbf{b}= \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ \end{vmatrix}.$$

Does the matrix whose determinant is this have any significance?

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It seems doubtful, given that it's only a matrix in a certain formal sense, with different elements in the first row than in the second and third rows. One might expect a matrix with further significance to correspond to some linear transformation, but for that it would have to be a "proper" matrix. – joriki Jun 5 '12 at 11:45
I agree with joriki that it is mainly just a helpful mnemonic device to remember how to compute the product, and not a real matrix. – rschwieb Jun 5 '12 at 11:57
up vote 2 down vote accepted

You can let the matrix act by ordinary matrix multiplication on ordinary vectors in three-dimensional space.

This will transform a vector in a triple containing the original vector and the lengths of the two projections on $a$ and $b$.

While I feel that this counts as "any significance", it isn't very satisfactory, because the matrix, as you presented it, does not allow for matrix multiplication.

I think it is a much more useful point of view, to first view $i$,$j$,$k$ as three scalar variables (better denoted by $x$, $y$, $z$), then take the determinant of your matrix and then regard the cross product as the gradient vector this determinant.

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