Is it true that $\left(\vec w-\dfrac12\vec u\right)\perp\left(2\vec w-\vec u\right)\;\forall\vec u,\vec w$?

Question

True or false? Let $\vec u$ and $\vec w$ two vectors. Then $\vec w-\dfrac12\vec u$ and $2\vec w-\vec u$ are perpendicular.

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I think is false. Let $\vec u=(1,0,0)$ and $\vec w=(0,1,0)$ two vectors. Then $$\left(\vec w-\dfrac12\vec u\right)\cdot\left(2\vec w-\vec u\right)=\left(-\dfrac 12,1,0\right)\cdot\left(-1,2,0\right)=\dfrac12+2=\dfrac52\neq0.$$

Is it correct?

Thanks!

Answer 1

They are parallel $$\left(2\vec w-\vec u\right)=2\left(\vec w-\dfrac12\vec u\right)$$ this shows one is a factor of the another.