Let $n$ be a nonnegative integer, let $k$ be a field, and let $a=(a_0,\ldots,a_n)$ and $b=(b_0,\ldots,b_n)$ be elements of $k^{n+1}\setminus\{0\}$. Then the homogeneous coordinate points $[a_0:\cdots:a_n]$ and $[b_0:\cdots:b_n]$ are equal in $\mathbb{P}^n_k$ if and only if there exists $\lambda \in k^\times$ such that $b_i=\lambda a_i$ for every integer $i$ with $0 \leq i \leq n$.