Interesting, I think it’s actually far more intuitive to think of it geometrically. I’m not sure what my brain is doing in order for this mental projection to help, but this is exactly how I made dot products “click” for me. I started to think of them in multidimensional space, almost physically (though in a very limited sense since my brain came from a monkey and generally fires on a couple cylinders).
I expect it’s like how learning to play by ear is more intuitive than sheet music. That’s great if you’re an amateur. If you’re dealing with tensors or somesuch trying to design a fusion reactor that’s probably a crutch.
This is a very odd statement and depicts the different ways human brain works. As a musician, I find playing (or thinking music in terms of) sheet music so much more intuitive than play by ear. It feels like the very reason people notate, write music is because anything written down is easier to think/play than anything listened.
I can intuit a lot of things about music and even visualize some of it, but eventually I hit limitations. What I learn through these intuitions still applies as my ability to mentally visualize or model the music begins to fail, though.
It's similar with vectors. Once you have the orchestral equivalent of vectors, there's no way I'm visualizing it and doing mental geometry. However, what I learned and the modelling I developed from the "casio keyboard playing jingles" equivalent of vectors is still useful and applicable.
I guess this is the point where playing by ear or mentally modelling things fails, and notation is far more helpful. Yet if a lot of us approach these complex works from the notation angle first, we might feel pretty lost and uncertain about what we're doing with it and why.
I can tell I'm not articulating this well, but I like the musical analogy and wanted to get that out.
One cannot intuitively think about higher than 3 dimensions. Even for most their intuition is often wrong in 3D space. It's quite accurate for 1D and 2D.
Richard Hamming has a whole section lecture to make everyone realize precisely this [1]. This was an eye opener to me.
Ehh… you can intuitively think about it. Intuition is something you develop with time as you gain familiarity with a subject. You just can’t bring all of your intuitions about 3D space into higher-dimensional spaces.
I took many classes in school where we worked with higher dimensional spaces. You wouldn’t send a physics major a lecture on physics, say it was “eye opening”, and expect them to feel the same way about it. It is stuff they have already seen before. Maybe their eyes are already open.
