cos⁡(θ(x,y))=⟨x,y⟩‖x‖2‖y‖2\cos(\theta(x,y)) = \frac{\langle x,y\rangle}{\Vert x \rVert_2 \lVert y \rVert_2}

with cosine similarity closer to 1 means more similar, farther from 1 means more different

Notes: can think of as natural ‘inverse’ to Euclidean distance ||𝐱−𝐲||22||\mathbf{x}-\mathbf{y}||_2^2

Suppose 𝐱,𝐲\mathbf{x},\mathbf{y} are unit vectors, then,

||𝐱−𝐲||22=⟨𝐱−𝐲,𝐱−𝐲⟩=||𝐱||22+||𝐲||22−2⟨x,y⟩=2−2cos⁡(θ(𝐱,𝐲)) ||\mathbf{x}-\mathbf{y}||_2^2 = \langle \mathbf{x} - \mathbf{y},\mathbf{x} - \mathbf{y} \rangle =||\mathbf{x}||_2^2+||\mathbf{y}||_2^2-2\langle x,y \rangle = 2 - 2 \cos(\theta(\mathbf{x},\mathbf{y}))