# Why vector lengths may not be preserved?

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

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

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

Staff Emeritus
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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.

Nusc
Not preserved under what?
length

Nusc
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

Mentor
2021 Award
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.

Nusc
I see. thank you