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

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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.

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

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

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

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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.

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lengthfresh_42 said:Not preserved under what?

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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_jOrodruin said:

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.

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

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I see. thank you

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.

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.

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.

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.

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.

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