Attention mechanisms are central to the success of large language models (LLMs), enabling them to capture intricate token dependencies and implicitly assign importance to each token. Recent studies have revealed the sink token, which receives disproportionately high attention despite their limited semantic role. In this paper, we first expand the relationship between the sink token and other tokens, moving beyond attention to explore their similarity in hidden states, considering the layer depth. We observe that as the layers get deeper, the cosine similarity between the normalized hidden states of the sink token and those of other tokens increases, and that the normalized hidden states of the sink token exhibit negligible changes. These imply that other tokens consistently are directed toward the sink token throughout the layers. Next, we propose a dynamic token selection method, called OrthoRank, using these findings to select important tokens. Specifically, in a certain layer, we define token importance by the speed at which the token moves toward the sink token. This is converted into orthogonality with the sink token, meaning that tokens that are more orthogonal to the sink token are assigned greater importance. Finally, through extensive experiments, we demonstrated that our method results in lower perplexity and higher zero-shot accuracy compared to layer pruning methods at the same sparsity ratio with comparable throughput, while also achieving superior performance on LongBench.