Q:

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.

A:

As ACuriousMind has already noted, you can geometrically interpret the length of the cross product of two vectors as the area of the parallelogram (or as twice the area of the triangle) spanned by them, and (the absolute values of) its components as the areas of the projections of that parallelogram onto the coordinate planes.
As for the dot product of two vectors, based on the law of cosines, you can interpret it as half the difference between the sum of their squares and the square of their difference:

$$\|\vec a - \vec b\|^2 = \|\vec a\|^2 + \|\vec b\|^2 - 2(\vec a \cdot \vec b).$$
In other words, taking the vectors to be two sides of a triangle, the dot product measures (half) the amount by which Pythagoras' law fails for this triangle.
Another way to geometrically interpret (the absolute value of) the dot product is as half the area of the triangle formed by rotating one of the vectors by 90° in their common plane, and then taking the resulting vectors as two sides of a triangle:

This follows from the well known dot product formula $\vec a \cdot \vec b = \|\vec a\| \|\vec b\| \cos \gamma$, where $\gamma$ is the angle between $\vec a$ and $\vec b$, from the triangle area formula $T = \frac12 \|\vec a'\| \|\vec b\| \sin \gamma'$, where $T$ is the area of the triangle formed by the vectors $\vec a'$ and $\vec b$ and $\gamma'$ is the angle between them, and the fact that the angles $\gamma$ and $\gamma'$ are complementary, and so $|\cos \gamma| = |\sin \gamma'|$.
Note the similarity with the cross product here.  In fact, we always have $\|\vec a \times \vec b\| = |\vec a' \cdot \vec b|$, where $\vec a'$ is $\vec a$ rotated by 90° in their common plane (or in any of the planes, if there are several)!

Ps. I did notice (after posting this answer) that you asked specifically about the units of the products and "not about geometric interpretations."  Even so, these examples should at least show that both the dot and the cross product of two length vectors can, in fact, be meaningfully interpreted as areas, and it should therefore not be surprising that, if the original vectors have units of, say, meters, then their product will be measured in square meters.

A:

The length of the cross product of two vectors is the area of the parallelogram spanned by them, so the square-meters are the correct unit as well as geometrically meaningful - it's really an area. The $x$-component is the area of the projection of the parallelogram onto the $y$-$z$-plane, the $y$-component the area of the projection onto the $z$-$x$-plane and the $z$-component is the area of the projection onto the $x$-$y$-plane.
The unit of the dot product is not really meaningful. It's by definition the length of the projection of the first vector onto the second times the length of the second (or vice versa), which does not straightforwardly correspond to any area. It gets units of square-meters by definition, but there is no deeper interpretation behind it I could see.

A:

It seems to me that you can always factor out the units of the components of a vector and rewrite it as a (physically) dimensionless vector times those units.  Then the dot and cross products work on the dimensionless vectors and the attached units simply multiply together as they do in any non-vector problem.
As far as interpreting the resulting units, it will depend on whether you have constructed something that has a physical meaning or not. I suppose you could take the cross product of two force vectors but I don't think it will mean anything (but I'd be interested to see an interpretation if someone can think of one).  Of course you can take the cross product of a position vector and a force vector and interpret the result as a torque.
Your dot product of two position vectors has square meters as units because it corresponds to a projected length times another length.  And the cross product does correspond to an area as others have already pointed out.

