alt Hacker News> that logic also suggests Strang's explanation is defective
I haven't read Strang's book, so I can't comment on that. But yeah, if it never mentions the formula ||a||·||b||·cos(θ) or at least talks about how the dot product relates to parallel and orthogonal vectors, I would consider that a big hole in an introduction to the dot product.
> How would that have fit with the goals of this post?
Because the post is titled "an introduction to linear algebra... the dot product", and this is something that I believe should be in anything that considers itself an introduction to the dot product.
You seem to disagree, and I'd like to ask: why? I think this a fundamental aspect of the dot product, again, just as fundamental as the relationship between complex multiplication and rotation. I think my view is common.
> calling out a blog post
I didn't intend to do anything as strong as "call out" the blog post. I just wanted to express surprise at someone so strongly praising ("its sauce is stronger [than 3B1B's video series]") an alright post.
Well, he's one of the most famous and best-respected educators of linear algebra, and this is an unusually basic piece of linear algebra to be taking aim at him for, so one answer is: if you have to ask whether his approach is defective, you should first evaluate how strong your own understanding is. That's argument from authority, but then: Strang is an authority. The typical push-pull on a message board is between Strang and Axler, and, if you want to find out if Axler is going to save your argument, flip to 6.A in LADR.
The direct response to your question is: the centrality of the angular interpretation of the inner product is the kind of thing I feel like you'd say if your primary purpose for learning linear algebra is to program video games. Linear algebra isn't "about" geometry, and, in particular, Strang's teaching goal centers vector spaces and the relationship between spaces. You need inner products to apply linear transformations with matrices. You basically don't need the angular interpretation... at all? if your goal is to teach linear algebra as a tool for data manipulation and, as Strang did in his last course, as a vehicle for deep learning.
(You need orthogonality, but you didn't say `a⟂b=0`, you said `|a||b|cosθ`).
I'm not rating this blog post "higher" than 3B1B; the comparison doesn't even make sense. The blog post and the video series simply have different objectives.