Let $X$ be a Banach space and let $x,y$ be two distinct vectors in $X$ with the following property:
$(1)$ $\|x\|=\|y\|=1$.
$(2)$ $x\neq \mu y$ for any unimodular scalar $\mu$.
From this can we assume that $x-\alpha y\neq \theta$ for any scalar $\alpha$, where $\theta$ denotes the zero vector in $X$?
My reasoning:
Suppose on the contrary that $x-\alpha y= \theta$ for some $\alpha$, then we have that $\|x\|=|\alpha|\|y\|$. Since $x$ and $y$ are unit vectors, we must have $|\alpha|=1$. Therefore, $x=\alpha y$ for some unimodular scalar $\alpha$, which is a contradiction. Thus, $x-\alpha y\neq \theta$ for any scalar $\alpha$, as expected.