Assume in a text we have two vectors v and z. Each vector contains m elements. Which one is correct:

1- " the vector v contains ..."

2- " vector v contains ..."

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    We'd need a full context (complete sentence, plus some indication of what text came before it) to give a definitive answer on this point. But for most purposes it's probably sufficient to say the article is optional in such constructions (but most people probably wouldn't include it). – FumbleFingers Reinstate Monica Jun 27 '16 at 17:36
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  • More context is needed. #1 would be used at the beginning or possibly middle of a sentence, and #2 might be used in the middle of a sentence. – Peter Jun 27 '16 at 18:29

You would, in most cases, not use "the" in this example. That is because "vector v" is a unique object of which there is not more that one. Because it is unique, it doesn't need articles to distinguish it from a crowd (that crowd doesn't exist). You can still use "the" if you wanted to, but most people would not. If the vector did not have a specific indicator (like the name "v" given to this one), then the articles would apply, as a vector can take on many different forms and properties.

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