What is the notation for the distance between two position vectors?

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Discussion Overview

The discussion revolves around the notation for the distance between two position vectors in three-dimensional space. Participants explore different ways to express this distance mathematically, including the use of norms and metrics.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant proposes using the equation r = ||(s - v)|| to represent the distance between two position vectors, where ||x|| denotes the modulus function.
  • Another participant confirms that ||x|| is a symbol used for the norm of x, noting that it must be defined.
  • Further clarification is provided that while the Euclidean metric is commonly used, other metrics can also be applied.
  • A participant lists several definitions of the norm, including the Euclidean norm, the Manhattan norm, and the maximum norm, suggesting that there are various ways to define distance.

Areas of Agreement / Disagreement

Participants generally agree that the notation for distance can be expressed using norms, but there is no consensus on which specific metric should be used, indicating multiple competing views.

Contextual Notes

The discussion highlights the need for clarity in defining the norm being used, as different definitions can lead to different interpretations of distance.

CraigH
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If I have two 3 dimensional position vectors:

s
and
v

What is the notation for the distance between them?

I know to calculate the distance I would use:

r = SQRT ( (s1-v1)^2 + (s2-v2)^2 + (s3-v3)^2 )
where s1 is the first element in the position vector s etc.

What is an equation that represents this? Can I use the equation:

r= ||( s - v )||

where ||x|| is the modulus function?
 
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Sure.
 
||x|| is symbol used for the norm (modulus) of x. It has to be defined. Usually the Euclidean metric is used, but others may be used.
 
mathman said:
It has to be defined. Usually the Euclidean metric is used, but others may be used.

Hi mathman,
Thanks for the answer. What do you mean when you say it has to be defined?
 
Let v = (x,y,z)

Definition 1: ||v|| = (x2 + y2 + z2)1/2
Definition 2: ||v|| = |x| + |y| + |z|
Definition 3: ||v|| = max(|x|,|y|,|z|)

There are endless possibilities, such as using nth power with nth root, analogous to Euclidean.
 

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