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If I draw 2 perpendicular line segments on the ground, 3 meters and 4 meters, how far into the sky does their cross-product extend?
What if instead the line lengths are 300 cm and 400 cm?
Can someone explain cross-products to me in a way that won't make Pythagoras angry?
Nice question! So you have found that the natural units for the cross-product are those of an area and not of a length. This is correct: the most natural way of thinking of the cross product of two vectors is to think that they measure the oriented area of the parallelogram they generate.
At a more advanced level, this generalise in a slightly different form to any number of dimensions to what is called a "volume form".
In particular cases, and 3-dimensional Euclidean space is among them, there is a way to build a vector out of this area element: To the surface spanned by the two vectors associate the vector perpendicular to it, with direction given by the right-hand rule and with magnitude given by the area of the parallelogram.
While we have recovered the description of the cross-product in terms of a vector and not of an area element, the fact that the appropriate units are those of an area is still there. In fact, either you are working with non-dimenisonal quantities, or the physical dimension of $a\wedge b$ is the product of the dimensions of $a$ and $b$. This generates no paradoxes: $12 m^2=12\cdot 10^4 cm ^2$. Measure the cross product with the units of a length and you will be in troubles: $12m\neq 12\cdot 10^4 cm$.
