I Are There Other Ways To Represent Vectors Besides Arrows?

mech-eng
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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?
 
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mech-eng said:
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.
 
mech-eng said:
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:$$\ket \alpha, \bra \beta$$
 
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fresh_42 said:
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.
 
mech-eng said:
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 ...

S%C3%BCtterlinschrift.png


... or Fraktur ##\vec{x}=\mathfrak{x}\, , \,\vec{y}=\mathfrak{y}\, , \,\vec{z}=\mathfrak{z}.##
 
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mech-eng said:
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)
 
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Look to differential forms and to clifford algebra notation too.
 
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mech-eng said:
However, I am aware of non-Euclidean geometries.
You're one step ahead of Euclid, then!
 
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jedishrfu said:
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
1664150252335.png


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

All of these are based on Schouten's work...
1664150534323.png
1664150581540.png

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

1664150967473.png


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