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I am currently studying linear algebra and one of the things I am having trouble with concerns understanding the questions being asked to me. The following question I am having trouble defining.

The following question is in R2 space.

If I take any three vectors, u, v, w in the plane, will there always be two different combinations that produce... (and the question continues, I understand the rest of the question)

What I don't understand is the "If I take any three vectors, u, v, w in the plane".

Does this mean I can take for example u, u and u (effectively three u vectors, as it states: any three vectors and figure out different combinations with them etc.). Or perhaps u, u and v? Or does it mean u, v, w specifically.

I know this might seem like a stupid question.

Karl Morrison
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4 Answers4

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Although one answer was already accepted, I'd like to emphasize that these "letters" are just "names". Imagine the question was

If I take any three vectors in the plane and call them $u$, $v$, $w$, will there always be...

In the question, it is not defined which vectors this are. You can take the vector $(1,2)$ and call it $u$. Then you can take the vector $(1,2)$ and call it $v$...

These names are just there so that the question can later refer to the vectors that you have chosen.

And, as pointed out in the other answers, there are no additional constraints mentioned for these vectors, so you are free to choose the same vector three times, and give it a different name each time.

(This may not add information over the existing answers. But the way how the question was written, particularly the question whether one could choose "u, u and u", sounded like a very basic misunderstanding. One could say: No, you may not choose "u, u and u", because the names that you may use are part of the question. So even if all three vectors are equal, you have to assign different names to them - otherwise, the remaining part of the question would not make sense...)

Marco13
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    I understand what you are saying and it is my fault for not including such info, in the question(in my book) the three vectors are defined. Just shows how important every little bit of info is! – Karl Morrison Oct 16 '14 at 17:26
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If they wanted you to not take $u,u,u$ or $u,u,v$, they would've said "any three distinct vectors". Another common and related (but a bit stricter) condition is "any three pairwise non-parallel vectors".

Arthur
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With any three we do mean any three, so if you'd like you could choose $u=v=w=(312,\frac{\pi}{e^7})$, otherwise the question should explicitly state that $u,v,w$ are different/distinct

konewka
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The result should hold for every possible choice of $\vec{u}, \vec{v}$, and $\vec{w}$. Equal or unequal. So if the answer is 'no', there should be a counterexample. If the answer is 'yes', there should be a general proof (valid for all possible choices of the three vectors).

paw88789
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