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# How to interpret the units of the dot or cross product of two vectors?

Suppose I have two vectors $$a=\left(1,2,3\right)$$ and $$b=\left(4,5,6\right)$$, both in meters.

If I take their dot product with the algebraic definition, I get this:

$$a \cdot b = 1\mathrm m \cdot 4\mathrm m + 2\mathrm m \cdot 5\mathrm m + 3\mathrm m \cdot 6\mathrm m = 4\mathrm m^2 + 10\mathrm m^2 + 18\mathrm m^2 = 32 \mathrm m^2$$

Dimensional analysis tells me that this is in meters squared, if I understand correctly.

Doing the cross product, however, I get this:

$$a \times b = \left[ \begin{array}{c} 2\mathrm m \cdot 6\mathrm m - 3\mathrm m \cdot 5\mathrm m\\ 3\mathrm m \cdot 4\mathrm m - 1\mathrm m \cdot 6\mathrm m\\ 1\mathrm m \cdot 5\mathrm m - 2\mathrm m \cdot 4\mathrm m\\ \end{array} \right] = \left[ \begin{array}{c} -3 \mathrm m^2\\ 6 \mathrm m^2\\ -3 \mathrm m^2\\ \end{array} \right]$$

This doesn't make sense to me either.

I don't know if I'm thinking about this in the right way, so my question is this: when dot or cross-multiplying two vectors, how do I interpret the units of the result? This question is not about geometric interpretations.