Will order it today! Thank you!

Funnily enough I hadn’t considered linear combinations yet. I agree they’re really useful, but how would somebody have thought to apply that definition to little arrows in space and noticing that they measure alignment?

Put another way, if I want to teach my Calc III class about these products, how could I have them “naturally” come out of some exploration, where they could make the “oh yeah” leap themselves?

I mean… yes I know quaternions work, but (1) that still doesn’t answer the question of “how would someone come up with this on their own”, and (2) it seems like even more of a stretch to do so, coming up with three different imaginary units with a particular relationship.

Oh yeah, and I know you get the geometric product if you decide i² = j² = 1 and ij = -ji are bivectors, and you get lots of nice generalizations.

But that would require you're comfortable a priori with the idea of adding two different-graded(?) objects together (scalars and bivectors).

But where I'm stuck is how someone would have decided on those choices for i², ij, ji, and j².

What would motivate them if you didn't already know to try them?

I vaguely smell something about what should happen when vectors are parallel vs. perpendicular, but I'm having trouble making it precise.

If you make certain choices about what i², ij, ji, and j² should be, you can then recover the other products:

* If i² = j² = 1 and ij = ji = 0, you get the dot product.
* If i² = j² = 0 and ij = –ji, you get the cross product if ij = k and the wedge product if ij is something new (a bivector).

Personally my first instinct would have been something like this:

If I have two vectors u = a i + b j and v = c i + d j, I would expect their product to "respect" distributivity:

uv = ac i² + ad ij + bc ji + bd j²

And if we stop here, if I understand right, this is the tensor product.

Personally I wouldn't round it at all without any further context. 😜

For context, this came up as I was searching for why anyone ever used to care about the normal line to a curve.

en.wikipedia.org/wiki/Evolute

Apparently the evolute of a curve is the envelope of all its normals!

First thought: "Who cares?"
Second thought: "Well, SOMEBODY used to care... but why?"

Okay yeah wow that’s pretty jam packed

Oh that’s interesting — somewhat related, can I DM you asking you some stuff about that kind of transfer stuff?

One of the reasons I’m asking this question is I’m working on a calculus textbook, and trying to decide on how to sequence it and what to include or leave out.

Okay since both of y’all are talking about UW I’m now curious.

What’s calculus like at UW? How jam packed is it? Do you per chance have syllabi or something to give an idea of what topics are covered?

Well normal *vectors* have a LOT more applications: the TNB frame, the relationship between the gradient and the level curves, and eventually surface normals for surface integrals.

I’m not convinced any of those speak to the utility of the normal *line*.

I need that recipe 👀

> Be me in 6th grade
> Beat final boss
> See these equations
> Have to beat final boss again
> Frantically scribble problems on paper
> Beat final boss again to finish writing
> Get calculator out
> It only holds 8 digits
> Work out answers by hand
> Enter password
> Incorrect password
> Repeat...