# Method to combine a vector quantity into one number.

• ashleykr145
In summary: Product of Inertia?In summary, Ashley is looking for a way to combine the magnitude and direction of a vector (in mm, and angle in degrees, although can be converted to radians) into one single figure, that will not repeat itself for all lengths up to 5 mm within 360 degrees. He has tried a couple of methods, but is now unsure as to whether this is mathematically possible. Additionally, Ashley has failed to find any literature that has attempted this.
ashleykr145
Dear Physics forum subscribers,

I currently have a problem whereby I need to combine the magnitude and direction of a vector (in my case a length in mm, and angle in degrees, although can be converted to radians) into one single figure, that will not repeat itself for all lengths up to 5 mm within 360 degrees. This is for use in Statistical analysis using correlation coefficients to test against one of my output variables.

I have tried a couple of methods, however am now unsure as to whether this is mathematically possible? Additionally I have failed to find any literature that has attempted this.

I've tried to keep this message as simple as possible, however if you need any more information as to methods I've tried or more information on my study please let me know.

I am a Sports Biomechanist and not a pure Physicist, so do excuse me if I have posted this within the wrong forum.

Ashley Richardson
PhD Student
University of Hertfordshire
UK

Good afternoon Ashley and welcome to Physics Forums.

This is a mathematical question and really belongs in the statistics section.
I suggest you contact one of the mentors and ask for it to be transferred so that a maths specialist might see it.

One ring to bind them all huh?

You have been watching too much Hobbit.

There may be such a mathematical object known as a complex number that can act as a single entity, but you will need to describe your statistics methods for someone to tell if it can be employed in this case.

go well

Complex numbers are a nice idea.

While it is possible to "compress" two real numbers into a single one in a reversible way, I do not think you want that. All those versions are ugly for a statistical analysis, and small variations in one variable can lead to large variations in the compressed version (or vice versa). In addition, it is problematic for a computer, which has a limited precision.

mfb said:
Complex numbers are a nice idea.

While it is possible to "compress" two real numbers into a single one in a reversible way, I do not think you want that. All those versions are ugly for a statistical analysis, and small variations in one variable can lead to large variations in the compressed version (or vice versa). In addition, it is problematic for a computer, which has a limited precision.

Hello mfb, I think that is a tad unfair.

There is an example from mechanics of just such a combination, but I can't tell if it is of use here without more information.

The example? Product of inertia.

A product is not injective. If I give you a product of two real numbers (like "5"), you have no way to find the two factors.
ashleykr145 said:
one single figure, that will not repeat itself for all lengths up to 5 mm within 360 degrees.

A product is not injective

So?

The product of inertia is one value that can distinguish between the two components in a way that the moment of inertia cannot.

It violates the ashleykr145's requirements.
If you multiply the values, 2mm and 90° gives the same product as 4mm and 45°.

Mmb, Do you want to help or just quibble?
Have you actually anything useful to offer tha Lady/Gent by way of welcome to PF?

The product of inertia (in 2D) is formed by an averaging process from quantities that have two components, just like vectors.

Maybe it can be adapted, maybe it can't.

But to suggest that I was talking about multiplying mm by degrees when I mentioned product of inertia, what were you thinking?

Could you clarify your suggestion? I think I don't understand it.
Is it an injective transformation [0,5] x [0,360] -> R? If not, where is the relation to the initial problem?

This thread explains about product of inertia and the averaging process, in particular posts 7 and 8 explain the significance of POI.

Unfortunately the formatting process has not aged well.

I wondered if the vectors could be plotted as complex numbers or just vectors and their ends regarded as mass points in some sort of POI scheme.

Alternatively I don't know enough about statistics to know if there are any complex stat functions that could be employed, hence my suggestions to move this to the statisticians.

Only Ashley can tell us how the statistics might be, but this is supposed to be Phd level, so it may be possible to work something out.

I don't see the relation to ashleykr145's problem to combine two real values in an injective way to a single one. My second question in post #9 was aimed at that.

Studiot said:
Despite the different mass distribution they have identical moments of inertia so the only way to distinguish is via the products of inertia.
This is just a result of your choice of the coordinate system. In a different system, I_xx and I_yy will be different for the two setups. And you can do a principal component analysis to get coordinate-independent values.

ashleykr145 said:
I currently have a problem whereby I need to combine the magnitude and direction of a vector (in my case a length in mm, and angle in degrees,

What you need to do here (in the forum) is explain the problem completely and tell what this vector represents and what other quantities are involved - otherwise we can only amuse ourselves by idle speculation about a satisfactory technique.

## 1. What is a vector quantity?

A vector quantity is a physical quantity that has both magnitude and direction. Examples of vector quantities include velocity, force, and displacement.

## 2. Why would you want to combine a vector quantity into one number?

Combining a vector quantity into one number allows for easier comparison and calculation of the overall magnitude of the quantity. It also simplifies the representation of the quantity in mathematical equations.

## 3. What is the method used to combine a vector quantity into one number?

The most common method used to combine a vector quantity into one number is vector addition. This involves adding the magnitudes of the vectors and considering their relative directions.

## 4. Can you give an example of combining a vector quantity into one number?

Yes, for example, if you have two forces acting on an object at right angles to each other, you can use vector addition to find the net force acting on the object by finding the square root of the sum of the squares of the individual forces.

## 5. Are there any limitations to using this method?

Yes, this method may not be applicable for vector quantities that have different units or are not in the same plane. It also does not take into account the direction of the resulting vector, only its magnitude.

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