# Meaning of each member being a unit vector

• I
• AlekM
In summary, the conversation discusses the meaning of each member of a set of observed directional data being a unit vector and how the products of each tensor can be averaged. There is confusion about the notation and what the values of n and i represent, with speculation that n represents a normalized vector representing a direction and i represents the axes that make up each vector. The paper being referenced is "Distribution of Directional Data and Fabric Tensors" and a thesis on fabric tensors is also mentioned as a useful source for further understanding.
AlekM
TL;DR Summary
Meaning of each member being a unit vector, and how the products of each tensor can be averaged.
Summary: Meaning of each member being a unit vector, and how the products of each tensor can be averaged.

Hello!
I am struggling with understanding the meaning of "each member is a unit vector":
Let n(1), n(2), and n(N) be observed directional data, where each member is a unit vector. The most fundamental quantities are various averages of them. Since we are trying to seek tensor quantities to characterize the data distribution, we first consider the average of their tensor product, or the “moment”, and put
Ni1i2...in = ⟨ni1ni2...nin

I can see that N would represent the number of samples, and the pointy bracket represents an average:
Average of ith component of a = ⟨ai⟩ = (ai1+ai2+aiN)/N

However the notation of the averaged tensors in the text is ni2...nin, which leads me to wonder what i and n represent. It states that each member (of n?) is a unit vector, does that mean that each value of n is a normalized vector representing a direction? Or could a single value of n contain multiple vectors of directional data?

The paper later states that they adopt the summation convention over tensor indices, would this mean that something like i represents the axes that make up each vector (such as x,y,z in Cartesian) and 1 through n represents the axes that make up each vector? In that case it would seem that the directional vectors from each sample are averaged together. I'm not sure if I am on the right track or entirely incorrect.

Thanks in advance for any help,
Alek

I read it as
AlekM said:
... does that mean that each value of n is a normalized vector representing a direction ...
We have ##N## observations and they span the phase space ##n^{(1)},\ldots,n^{(N)}##, i.e. each observed value ##n^{(i)}## is considered a basis vector of length ##1##. From there we can consider various other vectors in this space.

However, without context this is a bit of a guess.

AlekM
AlekM said:
The paper later states

Is the paper online? Can you give a link to it?

I just found that a page earlier direction is represented by unit vector n, so I think this is correct. However my next confusion is what nin represents.

As for the paper it is called "Distribution of Directional Data and Fabric Tensors," and I am referencing the beginning of page 2: https://www.sciencedirect.com/science/article/pii/0020722584900909

I can't read the paper since I'm not a member.

It surprised me how many articles are written about "fabric tensors". I hadn't heard of them before.

This thesis https://scholarworks.montana.edu/xm...2264/ShertzerR0811.pdf?sequence=1&isAllowed=y defines them on page 58 of the PDF , page 38 of the document. Perhaps some forum member who is an expert in tensors can explain them.

AlekM
Wow I'm not sure how I missed that source, it is incredibly useful. Thanks for sharing!

I didn't realize that link wasn't open to everyone, for anyone else reading this thread here is a mirror: http://einsteiniumstudios.com/fabric_tensors.pdf

## What does it mean for a vector to be a unit vector?

A unit vector is a vector that has a magnitude of 1 and points in a specific direction. It is often used to represent direction or orientation in a mathematical or physical context.

## How is a unit vector different from a regular vector?

A regular vector can have any magnitude and direction, while a unit vector has a magnitude of 1 and a specific direction. Unit vectors are also often used as a basis for describing other vectors.

## Why is it important for each member of a unit vector to have a magnitude of 1?

The magnitude of a vector represents its length or size. By having a magnitude of 1, unit vectors are standardized and can be easily compared and combined with other vectors in mathematical operations.

## What is the significance of each member of a unit vector representing a different dimension?

Unit vectors are often used in multi-dimensional spaces to represent different axes or directions. Each member of a unit vector represents a different dimension, allowing for a more comprehensive description of a vector's direction and orientation.

## How are unit vectors used in physics and engineering?

Unit vectors are used in physics and engineering to represent direction and orientation in mathematical models and calculations. They are also used in vector calculus to describe the gradient and direction of a field. In engineering, unit vectors are used to describe the direction of forces and motion in mechanical systems.

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