A scalar measure of

vector similarity,

defined in a vector space \(\mathbb{R}^m\) with an orthonormal basis:

• algebraically,

  in terms of the matrix multiplication, and so the linear combination, as

  \begin{equation*} \vec{u} \cdot \vec{v} = \vec{u}^\mathsf{T} \vec{v} = \sum_{i = 1}^{m} u_i v_ i, \end{equation*}

• geometrically,

  in terms of the vector length, as

  \begin{equation*} \vec{u} \cdot \vec{v} = \|\vec{u}\| \|\vec{v}\| \cos \theta, \end{equation*}

  where \(\theta\) is the angle between \(\vec{u}\) and \(\vec{v}\).

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• Synonyms
• Intuition
• Properties
• Dimensionality: Vectors
• Dimensionality: Tensors
• Implement