We have the vectors $a =3u-2v$ , $b=-2u+v$ and $ c=7u-4v$. Prove that they are complanar.
Complanarity of three given vectors
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3Sure, $a-2b=c$. Was that really so hard? – 2012-11-16
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0The vectors u and v are NOT parallel. – 2012-11-16
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1Yes, I am AWARE of that. You asked to prove that they are complanar [sic.]. – 2012-11-16
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0I actually thought about doing it with the cross products,I know its not hard.. – 2012-11-16
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2copper.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. – 2012-11-16
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0On the contrary, thanks for explaining! – 2012-11-16
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1No,I still find it funny :) – 2012-11-16
3 Answers
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 $span\{a,b,c\}\equiv span\{u,v\}$.
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.
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.