# Why vector lengths may not be preserved?

• I
• Nusc
In summary, The length of a vector is not preserved when there is more than one metric on a vector space, as the existence of a metric is necessary to speak of length. Two different metrics on the same space may or may not be equivalent.

#### Nusc

Given some metric, what is an example where the length of a vector is not preserved?

I presume it's because a metric could have a scalar and that would not preserve the length.

Nusc said:
Given some metric, what is an example where the length of a vector is not preserved?
Not preserved under what?

Your question is rather ill defined. Preserved when exactly?

It is also not clear what you would mean by a ”metric having a scalar”. A metric is a type (0,2) tensor, not a scalar.

fresh_42 said:
Not preserved under what?
length

Orodruin said:
Your question is rather ill defined. Preserved when exactly?

It is also not clear what you would mean by a ”metric having a scalar”. A metric is a type (0,2) tensor, not a scalar.
if you define a metric g' = k g where k is some scalar function. you can expresss the lengths of a' = g' a_i a_j = k g a_i a_j

You need a metric in order to speak of length, not the other way around. If you have two different metrics, they might be equivalent ##g'=k\cdot g##, or not. The question about length is directly coupled to the existence of a metric - one metric. As an example you could consider the distance between two real numbers as ##|x-y|## or look at the discrete metric ##d(x,y)=1## as soon as ##x\neq y##. These are two different, non-equivalent metrics on one space, ##\mathbb{R}## in this case.

So as long as you only have one metric on a vectorspace, you cannot have two lengths.

I see. thank you

## 1. Why do vector lengths change when rotated?

Vector lengths change when rotated because the direction of the vector also changes. When a vector is rotated, its components are changed, resulting in a new length.

## 2. What factors affect the preservation of vector lengths?

The preservation of vector lengths can be affected by the magnitude and direction of the rotation, as well as the initial length of the vector.

## 3. Can vector lengths be preserved under any circumstances?

No, vector lengths cannot be preserved under all circumstances. They can only be preserved if the vector is rotated by an angle of 0 degrees or if the vector is rotated by a multiple of 360 degrees.

## 4. How does the direction of rotation affect vector lengths?

The direction of rotation can affect vector lengths because vectors have a direction, and when they are rotated, their direction changes as well. This change in direction can result in a change in the vector's components, and therefore its length.

## 5. Is there a way to calculate the new length of a vector after rotation?

Yes, the new length of a vector after rotation can be calculated using trigonometric functions such as sine and cosine. These functions can help determine the new components of the vector, which can then be used to calculate the new length.