I had a math graduate student teaching my linear algebra class. He taught dot and cross products entirely algebraically, never drawing vectors as arrows, but as arrays of numbers. When I suggested after class that teaching the visual representation might help some students, he pushed back. Visual understanding, he explained, was a crutch best avoided, because visual intuition could break down in higher dimensions. I thought that was a surprising perspective from a math graduate student, of all people.
Seems like an odd choice when talking about the cross product, since the cross product is only a thing in 3D. You can define analogous things in other dimensions but it becomes clearer and clearer that it’s not meaningfully a ‘product’.
So it doesn’t matter if your visual intuition for a cross product breaks down in higher dimensions - a cross product is only a thing in three.
This is very off topic, but the wedge product absolutely is "meaningfully a product", generalizes fine to arbitrary dimension, and has a perfectly reasonable visual/spatial/geometric interpretation.
(Indeed, we should entirely scrap the cross product in undergraduate level technical instruction and replace it with the wedge product; one happy effect will be replacing students' misleading spatial intuitions with better ones.)
Good point; you're right. I might be misremembering the cross product. I do remember that he didn't even teach the geometric interpretation of vector addition.
This is quite true, especially true when talking about direction (gradient) in high dimensional space. I don't think this can be avoided, since after all we are creatures living in 3D space where left right up down are quite well-defined, just need to make a mental note every time you have to deal with more than 3 dimensions.
I can see where he's coming from.
Geometric intuition is an useful tool but in this semester's linear 2 class I developed more because I stopped using it. It's too strong of a tool and blots out "dryer" intuition and methods, and also as you progress you find more and more places where it's not useful.
What's the geometrical intuition for whether two circles intersect in Q^2? who knows?
