Multiple dimensional vectors: Difference between revisions
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If there are more than three dimensions then we just sum from over more indices. That is the beauty of the sum notation for vectors. For example: |
If there are more than three dimensions then we just sum from over more indices. That is the beauty of the sum notation for vectors. For example if we have n dimensions, numbered from 1 to n: |
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<math>\vec \bold v = \sum_{k=1}^n v_k \vec \bold a_k </math> |
<math>\vec \bold v = \sum_{k=1}^n v_k \vec \bold a_k </math> |
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or when there are a countably infinite number of dimensions |
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or even |
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<math>\vec \bold w = \sum_{k=- \infty}^\infty v_k \vec \bold a_k </math> |
<math>\vec \bold w = \sum_{k=- \infty}^\infty v_k \vec \bold a_k </math>. |
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If there are an uncountably infinite number of dimensions, we move into the area of functions, and the sum must be represented with an integral as discussed [[Orthogonal functions#Functions and vectors, an analogy|here]]. |
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Latest revision as of 17:40, 26 September 2004
If there are more than three dimensions then we just sum from over more indices. That is the beauty of the sum notation for vectors. For example if we have n dimensions, numbered from 1 to n:
or when there are a countably infinite number of dimensions
.
If there are an uncountably infinite number of dimensions, we move into the area of functions, and the sum must be represented with an integral as discussed here.