 # True or false? Let vec(u) and vec(w) two vectors. Then vec(w) −1/2 vec(u) and 2 vec(w) − vec(u⃗) are perpendicular. traffig75 2022-09-17 Answered
Is it true that $\left(\stackrel{\to }{w}-\frac{1}{2}\stackrel{\to }{u}\right)\perp \left(2\stackrel{\to }{w}-\stackrel{\to }{u}\right)\phantom{\rule{thickmathspace}{0ex}}\mathrm{\forall }\stackrel{\to }{u},\stackrel{\to }{w}$ ?
I think is false. Let $\stackrel{\to }{u}=\left(1,0,0\right)$ and $\stackrel{\to }{w}=\left(0,1,0\right)$ two vectors. Then
$\left(\stackrel{\to }{w}-\frac{1}{2}\stackrel{\to }{u}\right)\cdot \left(2\stackrel{\to }{w}-\stackrel{\to }{u}\right)=\left(-\frac{1}{2},1,0\right)\cdot \left(-1,2,0\right)=\frac{1}{2}+2=\frac{5}{2}\ne 0.$
Is it correct?
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They are parallel
$\left(2\stackrel{\to }{w}-\stackrel{\to }{u}\right)=2\left(\stackrel{\to }{w}-\frac{1}{2}\stackrel{\to }{u}\right)$
this shows one is a factor of the another.