Lec 1: Dot product | MIT 18.02 Multivariable Calculus, Fall 2007
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange answerers point people to it as the starting point of a free, full multivariable calculus course, and askers cite its dot product derivation.
Recommended by 3 people across 3 different months, 2013 to 2018.
What it is
The first lecture of MIT 18.02 Multivariable Calculus (Fall 2007), taught by Prof. Denis Auroux and posted by MIT OpenCourseWare. It introduces vectors in 3D and the dot product, including its relation to the angle between vectors and the law of cosines.
What people said
Recommends math · 2013-09 · Mathematics Stack Exchange · on “Calculus books And some advice”
If you wanna learn some single variable calculus, MIT has this [LINK] series of youtube lectures. You can find multivariable calculus lectures of MIT here [this video] . Have fun!
Recommends The very fluffy Panda · 2014-04 · Mathematics Stack Exchange · on “Jumping back into Calculus III”
Calculus Early Transcendentals by James Stewart , I think it is one of the best books out there. Apart from Khan Academy on Youtube, you have MIT Multivariable Calculus [this video] and UCBerkely [LINK] .
Recommends Nishant · 2018-03 · Mathematics Stack Exchange · on “Where can I find introductory video lectures about calculus and analysis?”
MIT's OCW has a lecture called Highlights Of Calculus [LINK] by Gilbert Strang. You can also watch 3blue1brown's Essense Of Calculus [LINK] . These are the courses that give you a good feel of Calculus without the rigor of usual math classes. Here are some other playlists from MIT's OCW: Single Variable Calculus [LINK] Multivariable Calculus [this video] Single Variable Calculus by Herbert Gross [LINK] Multivariable Calculus by Herbert Gross [LINK]
Cites Detached Laconian · 2013-02 · Mathematics Stack Exchange · on “Why do we draw the $xyz$ coordinate system like this?”
Thanks a lot for the input! (I've just spent hours reading these wikipedia pages, imagining and drawing stuff. Feels like my intuition is getting somewhat better.) As for my Calculus teacher, he's certainly not the only one, e.g.: this [LINK] and this [this video] . I've never seen anyone draw it differently, to be honest.
Cites Jason · 2015-10 · Mathematics Stack Exchange · on “Law of cosines is equivalent to a dot product identity”
In this lecture [this video] , starting at around 18:00, it's shown the dot product identity $$a^T b=||a||_2 ||b||_2 \cos \theta$$ where $\theta$ is the angle formed b/w $a,b\in \mathbb{R}^3$, is the same as the law of cosines: $$||a-b||_2^2=||a||_2^2+||b||_2^2-2||a||_2 ||b||_2 \cos \theta$$ Prof Auroux shows they are equivalent: as he says, if you believe one, then you have a pf of the other. But I'm curious: what is the easiest way to prove one of these w/o relying on the other?