Dot product

A scalar measure of

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 \(\vec{v}\).

• Intuition
• Properties
• Implement