How can I show that the elements of the unitary sphere are not parallel?

Question

Let $(X,\|\cdot\|)$ be a normed vector space over the field $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$ and let the unitary sphere defined as $S_X:=\{x\in X;\|x\|=1\}$. If $u, v\in S_X$, with $u\neq v$, then $u$ and $v$ are not parallel. I just showed for $\mathbb{K}=\mathbb{R}$ but I need help for the case $\mathbb{K}=\mathbb{C}$.

Answer 1

It is false in both cases. If $\|x\|=1$ then $x$ and $-x$ are parallel though $x \neq -x$.