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Nusc
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Given some metric, what is an example where the length of a vector is not preserved?
Not preserved under what?Nusc said:Given some metric, what is an example where the length of a vector is not preserved?
lengthfresh_42 said:Not preserved under what?
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: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.
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.