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We have the vectors $a =3u-2v$ , $b=-2u+v$ and $ c=7u-4v$. Prove that they are complanar.

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Sure, $a-2b=c$. Was that really so hard? – copper.hat Nov 16 '12 at 7:06
The vectors u and v are NOT parallel. – xdfg Nov 16 '12 at 7:17
Yes, I am AWARE of that. You asked to prove that they are complanar [sic.]. – copper.hat Nov 16 '12 at 7:19
copper.hat is "complaining" because you asked to prove they are "complanar". It's a joke based on your typo. After my comment, there is now zero chance of it being funny. – Austin Mohr Nov 16 '12 at 7:26
No,I still find it funny :) – xdfg Nov 16 '12 at 7:32
up vote 0 down vote accepted

Without calculations, using a little theory. $a,b,c\in span\{u,v\}$. Consequently, $span\{a,b,c\}\subseteq span\{u,v\}$. Since $a\nparallel b$, (you said that $u\nparallel v$), it follows $span\{a,b,c\}\equiv span\{u,v\}$.

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Vectors $u,v$, not being parallel, span a plane. That plane contains $a$, $b$, and $c$. Hence these three vectors (and in fact all five) are coplanar by the definition of coplanar.

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It will suffice to show that the vector space spanned by $a,b,c$ is a plane, for then they all live on that plane. Note that $a$ and $b$ are linearly independent, so the dimension of the vector space they span is $2$. But $a-2b = c$, so $c$ already lives on the vector space spanned by $a,b$, and hence the vector space spanned by $a,b,c$ is $2$-dimensional, and so a plane.

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