Are There Other Ways To Represent Vectors Besides Arrows?

In summary, there are multiple ways to represent vectors, including geometrically as arrows, algebraically with mathematical symbols, and using different notations such as Dirac, Sütterlin, and Fraktur. Additionally, vector notation can be seen in the forms of differential forms and clifford algebra. Non-Euclidean geometries also offer different ways to represent vectors. These methods are all based on the work of mathematicians such as Schouten and Ricci.
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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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  • #2
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.
 
  • #3
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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  • #4
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.
 
  • #5
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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  • #6
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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  • #7
Look to differential forms and to clifford algebra notation too.
 
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  • #9
mech-eng said:
However, I am aware of non-Euclidean geometries.
You're one step ahead of Euclid, then!
 
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  • #10
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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FAQ: Are There Other Ways To Represent Vectors Besides Arrows?

1. What is a vector?

A vector is a mathematical quantity that has both magnitude and direction. It can be represented graphically as an arrow, with the length of the arrow representing the magnitude and the direction of the arrow representing the direction.

2. How are vectors represented mathematically?

Vectors are typically represented using a coordinate system, such as Cartesian coordinates, where the vector is described by its components in each direction. For example, a vector in 2-dimensional space can be represented as (x, y).

3. What is the difference between a vector and a scalar?

A vector has both magnitude and direction, while a scalar only has magnitude. For example, velocity is a vector quantity because it has both speed (magnitude) and direction, while temperature is a scalar quantity because it only has magnitude.

4. How are vectors added and subtracted?

To add or subtract vectors, their components in each direction are added or subtracted respectively. For example, to add two 2-dimensional vectors (x1, y1) and (x2, y2), their components are added to get the resulting vector (x1+x2, y1+y2).

5. What is the importance of vector representation in science?

Vectors are used extensively in science to describe physical quantities such as velocity, force, and electric and magnetic fields. They allow us to represent and manipulate these quantities mathematically, making it easier to solve complex problems and make predictions about the behavior of physical systems.

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