What Is a Vector? It depends! Geometric perspective vs Linear Algebra vs ℝⁿ

[@skeptical_Inquirer200]

What seems to be missing, to make this rigorous, is a definition of how two vectors are equivalent if they have different tails. Parallel also needs to be defined.

[@MathTheBeautiful]

For simplicity, I do not allow my vectors to have different tails. Parallel is defined the same way Euclid did it.

[@nbooth]

​@@MathTheBeautifulAssuming we already know what a point is from geometry, my first definition of a vector would be that it is the difference between two points. That is *almost* the same thing as a directed line segment, but I think it is a better definition. Even in one dimension, there is a difference between points and vectors, e.g. temperatures measured in °C or °F are points, but temperature differences are vectors. Same thing with points in time, which are points (as far as I know there is no time scale with an absolute zero), but timespans are vectors. The difference of points definition automatically handles "different tails", as the difference between points is invariant to translation. And it plays nicely with homogeneous coordinates where vectors and points are distinguished with zero or nonzero (usually 1) homogeneous coordinate respectively.

[@ReginaldCarey]

You can’t introduce dU without complex numbers. Similarly you can’t claim scalar vector multiplication scales the magnitude but then glibly introduce scaling by a negative number as reversing the direction. If a vector is a magnitude and direction, dU must represent a change in magnitude and a change in direction. -1 is a really bad representation of i^2.

[@ReginaldCarey]

Of course I forgot the \theta and the 2n in my definition of -1.

[@MathTheBeautiful]

Thank you for your interesting opinion!

[@ReginaldCarey]

I hope I’m not misspeaking. But aren’t complex numbers needed to describe vector vector multiplication?

[@MathTheBeautiful]

Are you referring to the scalar product? In that case, one can consider vector spaces over complex numbers, e.g. ℂ⁻ⁿ and complex-valued inner product. But complex numbers do not apply to geometric vectors as I'm not aware of a way to multiply a geometric vector by a complex number.

[@ReginaldCarey]

@@MathTheBeautiful Geometric Algebra defines vector vector multiplication than results in a geometric object that in 2D maps to the complex numbers. It combines the dot and wedge product, where the dot product is even and wedge product is odd. The wedge product includes an oriented unit multi vector and corresponds with the imaginary component of a complex number. The dot product corresponds with the real component of a complex number.

[@SkeletalBasis]

Quick request: maybe you could do a side video reconciling the notion that geometric vectors all proceed from one arbitrarily chosen point (on the one hand) with the tip-to-tail rule (on the other), which requires that they be all over the place? It looks to me that this would require a not entirely trivial construction involving equivalence classes of lines or rays, plus whatever else it takes to define 'sense' in the directional sense. One point of interest is that this would illustrate what it takes to advance from intuition to consistency --- a central issue in almost all kinds of thought.

[@MathTheBeautiful]

Good comment! To be honest, I don't see much of an issue. Tip-to-tail is just a set of instructions for constructing the sum and it doesn't apply that vectors can float around the space. But even if you do allow vectors to float around the space - yes, you can introduce an equivalence class where two vectors are equivalent if they are related by a parallel shift. Then you're back to having a simple vector space. The only reason I avoid considering that is that I err on the side of less.

[@nbooth]

As I posted elsewhere in the comments for this video, I think the construction is cleaner and easier if you distinguish between points and vectors, with vectors being considered the difference between two points. A point P and the vector P - O are different things. You can't add points, but you can add vectors to points and vectors to vectors. We have this distinction even with what we consider ordinary scalars, like temperatures in °F, which are points, but temperature differences are vectors. Even here it makes no sense to add points together, but you can add vectors to points and vectors to vectors.

[@AdrianBoyko]

I’d say that geometric vectors have a length and direction, and that’s it. They don’t have a position. Scalars are also added “tip to tail” But nobody says that when calculating 7+2 it is important to realize that 2 is at position 7!

[@ChaineYTXF]

Professor, did you study in France or in a French speaking country? Or have some French colleagues? Because "sens" and "direction" is a distinction I never hear in English courses! Only in France! Superb channel. Long time fan here

[@MathTheBeautiful]

No, I studied in Russia where we only had one word "direction". I discovered the word "sense" in a book I recently read, but I don't remember which. If I rediscover it, I'm come back here and share it.

[@TomSkinner]

I certainly was familiar with sense vs direction in my US education. But I'm ancient 😂.

[@AdrianBoyko]

Does the zero vector have “nonsense”? 🤔

[@MathTheBeautiful]

@@AdrianBoyko Haha - well done!

[@freddupont3597]

You are mentioning homework and exercises in the intro of the lesson; is there a link to them?

[@MathTheBeautiful]

When it's ready (enough), it'll appear on my website grinfeld.org

[@freddupont3597]

@@MathTheBeautifulThank you!

[@juliavixen176]

Unrelated to the content of this video, you need to turn off the auto focus of your video camera. Neither the camera nor the whiteboard are moving, so set the focus once at the beginning (verify it's in focus), and lock it so the camera will not change the focus of the lens.

[@MathTheBeautiful]

I appreciate the advice! Do you know if there's a way to put the camera in focus before turning AF off?

[@ReginaldCarey]

The full math is revealed when you stop restricting yourself to real scalars.

[@MathTheBeautiful]

Thank you for your comment!

[@Pluralist]