I appreciate how you put this subject. Your videos give me better explanations for Linear Algebra and Calculus 3. Thank you so much!
haydeeholliday
DR Pavel. You remark that geometric diagrams are not a demostration of vectorial addition properties, and I agree, but if you would like to have a formal demostration, where can I find them? . All books use the paralelogram rule and no one use a formal demostration. Thanks in advance
gguevaramu
02:35 Sir, what do you mean by formal proof ? Is there any other proof? If yes, please tell where I could get it.
kaminipandey
why the definition is like that for the length? I get the direction of the result of addition but the definition for its length isn't clear. despite the fact it will be clear in an argument includes components.
abdulrahmanmahmoudanter
The "Bourbaki" definition of a vector is that it's an element of a vector space. We can draw arrows in the plane or in space, and impose rules on how they combine, but it's not clear how to characterize these "geometric entities" other than saying that they live in a vector space. Having been brought up in this "Bourbaki" tradition, where a rigorous definition of an object is given by a set with a structure imposed on it, it's very difficult to read older works, like Cartan's Geometry of Riemannian Spaces, in that he doesn't define a structure (worse, he doesn't even define what a vector is!) before introducing an object, and distinguishes between a vector v, and the Cartesian triple that represents it after an imposition of Cartesian coordinates. It's almost as if he thought of vectors as "actual objects living in space". How to reconcile these two views?
maxisjaisi
You make maths seem interesting, thank you. Im in last year of maths and i was feeling a bit discouraged but not anymore thanks to you :)
darkdevil
Glad to see you're still doing this, and have almost 25k subscribers. Awesome. Recommending this to my MVC professor tomorrow.
chrstfer
thanks for all the efforts .I heard some people talking about attached or sticked vectors (don't have the freedom to move their application or first point) and free vectors. please have you got some more information, thanks again...
SkanderTALEBHACINE
(6:43) It does convince me since there is no stretching or rotation going on while you parallel transport a vectors in an Euclidean space, i.e. the sum will always become the greatest "hypotenuse" in the projected parallelepiped (i.e. all possible tip-to-toe configurations), therefore *a+b+c* must always be associative (in Euclidean space). It also follows from this that the order in how you add them does not affect the end result, i.e. *a+b+c* = *c+b+a* etc. Which is another way to say that vectors -always- _commute_ under addition.
A note about multiplying a vector with a number (scalar): in the physical interpretation, multiplying with a a scalar does not change the vector to another object. It just change the characteristic, i.e. magnitude, of the object. One can think about the scalar as "energy" transferred to, or removed from, the object, and as such we also realize that energy does not have any intrinsic dimension but its manifestation depends on the object itself, i.e. pressure, momentum, stress, etc.
