Yep, totally agree with the grouping – it shows the symmetry by thinking of a onto b, or b onto a. And yep, with line integrals, you usually have a defined Force and Path vector (it’s not as intuitive to project the Path onto the Force).
I love breaking down complicated ideas into easy-to-digest parts… yep, the integral is a wrapper around the idea of “applying” something piece by piece (of which multiplication and repeated addition are very simple base cases).
On the vector notation: I guess it depends on how deep you want to go :). For me, I take it as an axiom that “we want to figure out how much one vector pushes in the direction of the other, it will be a single number (called the dot product) and here’s why it should be what it is (i.e., x and y components interacting)”
Yes, relative direction is a good way to put it – the absolute doesn’t matter. In a system of relative coordinates (just angles between them), what do we know about the force of one onto the other? (Interestingly, I stumbled upon the concept of coordinate-free geometry today).
When we care about absolute direction, yep, we want different results when vectors that are 90 degrees apart but oriented differently are combined. I think we need to be careful with complex numbers though (for terminology) since the typical dot product gives 0 when perpendicular, but complex numbers do a rotation.
I might tweak it to “the dot product is multiplication, taking the difference in direction into account”. (I’ll have to tweak it). WIth all of these 1-liners, it’s a tradeoff between an immediate click vs. getting into too much detail.
And the feeling for these discussions is mutual, I love exploring these ideas!
@Joe: Thanks Joe. I agree, there’s an interplay between theory and application (I learned vector calc before E&M, and having physics examples definitely solidified my earlier understanding). I don’t think you can teach in some pyramid where the base ideas are learned, divorced from examples, and then the applications are sprinkled in later.
Appreciate the encouragement, I’d love to do the cross product & determinant in the future! (Two concepts I really, really want to get an intuition for & not just the mechanical calculation).