Are There Other Ways To Represent Vectors Besides Arrows?

[mech-eng]

TL;DR Summary

Looking for the different vector representations

Hello. I wonder about the representation of vectors, so I wanted to ask: how many different ways vectors be represented?

As far as I know two: Geometrically and Algebraically.

There is only one way to represent vectors in geometrically: arrows, however there are several or more methods to represent vectors algebraically but they are basically one dimensional of numbers. Sometimes there are mathematical symbols between the elements of the list as in 2i + 3j - 4k.

But are those all?

Can another tools be used instead of arrows for geometrical representation?

In more than 3d-space, can there be other ways?

 

[fresh_42]

Summary: Looking for the different vector representations

Hello. I wonder about the representation of vectors, so I wanted to ask: how many different ways vectors be represented?

As far as I know two: Geometrically and Algebraically.

There is only one way to represent vectors in geometrically: arrows, however there are several or more methods to represent vectors algebraically but they are basically one dimensional of numbers. Sometimes there are mathematical symbols between the elements of the list as in 2i + 3j - 4k.

But are those all?

Can another tools be used instead of arrows for geometrical representation?

In more than 3d-space, can there be other ways?

How do you define "representation"?

Your questions have so many hidden assumptions that there cannot be a serious answer.

 

[PeroK]

Summary: Looking for the different vector representations

Hello. I wonder about the representation of vectors, so I wanted to ask: how many different ways vectors be represented?

As far as I know two: Geometrically and Algebraically.

There is only one way to represent vectors in geometrically: arrows, however there are several or more methods to represent vectors algebraically but they are basically one dimensional of numbers. Sometimes there are mathematical symbols between the elements of the list as in 2i + 3j - 4k.

But are those all?

Can another tools be used instead of arrows for geometrical representation?

In more than 3d-space, can there be other ways?

There's Dirac notation, with its bras and kets:$$\ket \alpha, \bra \beta$$

 

[mech-eng]

How do you define "representation"?

Your questions have so many hidden assumptions that there cannot be a serious answer.

Sorry for being vauge and poor statements. Here representation is "notation" or how we can show vectors on paper or in a digital medium, in a program such as Octave.

 

[fresh_42]

Sorry for being vauge and poor statements. Here representation is "notation" or how we can show vectors on paper or in a digital medium, in a program such as Octave.

Oh, I thought you were asking how to draw vectors that can be quite troublesome if they are functions, the vector space is infinite-dimensional, or the field isn't of characteristic zero.

My teachers often used Sütterlin ...

... or Fraktur $\vec{x}=\mathfrak{x}\, , \,\vec{y}=\mathfrak{y}\, , \,\vec{z}=\mathfrak{z}.$

 

[berkeman]

Summary: Looking for the different vector representations

Hello. I wonder about the representation of vectors, so I wanted to ask: how many different ways vectors be represented?

Maybe have a read through this Wikipedia artlcle and follow some of the References to get more ideas:

https://en.wikipedia.org/wiki/Vector_(mathematics_and_physics)

 

[jedishrfu]

Look to differential forms and to clifford algebra notation too.

 

[mech-eng]

Maybe have a read through this Wikipedia artlcle and follow some of the References to get more ideas:

https://en.wikipedia.org/wiki/Vector_(mathematics_and_physics)

Here are more detailed explanations for the vector forms

https://en.wikipedia.org/wiki/Vector_notation

But I have never seen rectangular vectors in use;nor non-Euclidean vectors. However, I am aware of non-Euclidean geometries.

 

[PeroK]

However, I am aware of non-Euclidean geometries.

You're one step ahead of Euclid, then!

 

[robphy]

Look to differential forms and to clifford algebra notation too.

e.g. Ch 16 of Burke's unfortunately unfinished Div Grad and Curl are dead (1995)
https://people.ucsc.edu/~rmont/papers/Burke_DivGradCurl.pdf

This is a simplified version of figures seen in Misner Thorne Wheeler's Gravitation (1973).

All of these are based on Schouten's work...

from pages 15 and 22 of
Ricci Calculus [Der Ricci-Kalkül (1924)]
https://gdz.sub.uni-goettingen.de/id/PPN373339186
and

from p. 55 of Tensor Analysis for Physicists (1951)
https://www.amazon.com/dp/0486655822/?tag=pfamazon01-20