[ajkjk]

There are two ways to see every operation: a mathematical way and a physical way. The mathematical view of the dot product is an operation on vectors that adds their multiplied components, a·b = a_x b_x + a_y b_y + a_z b_z. The physical view of the dot product is what you said, comparing two vectors for similarity, or, in alternatively, multiplying their parallel components like scalars. The difference between these perspectives is in what is regarded as the defining property of the operation, which affects what you keep "fixed" as you vary aspects of the theory you're working in. For instance, when switching to spherical coordinates, the mathematical version of the dot product could still look the same, but the physical version has to change to preserve the underlying concept, which means its form becomes quite messy: (a_r, a_θ, a_φ)·(b_r, b_θ, b_φ) = a_r b_r (sin (a_θ) sin (b_θ) cos (a_φ - b_φ) + cos a_θ cos b_θ.

The difference in pedagogy seems to be which of these perspectives is treated as fundamental. Math education tends to treat the mathematical operations as fundamental. Physics treats the concept as fundamental and regards the operation as an implementation detail. It is very similar to how in software development you (for the most part) treat an API's interface as more fundamental than its implementation.

Unfortunately even physics books don't go over the intuition for underlying math very well, to their detriment. They seem to just assume everyone already perfectly understands multivariable calculus and linear algebra. I think it's because by the time you've gotten through a physics PhD you have to be completely fluent in those and the authors forget what it was like to find them confusing.