Dot Product

9. Dot Product

The dot product (or scalar product) of two vectors of the same dimension returns a scalar — the sum of componentwise products.

\vec{𝑎} \cdot \vec{𝑏} = [\begin{matrix} 𝑎_{1} \\ 𝑎_{2} \\ ⋮ \\ 𝑎_{𝑛} \end{matrix}] \cdot [\begin{matrix} 𝑏_{1} \\ 𝑏_{2} \\ ⋮ \\ 𝑏_{𝑛} \end{matrix}] = 𝑎_{1} 𝑏_{1} + 𝑎_{2} 𝑏_{2} + \dots + 𝑎_{𝑛} 𝑏_{𝑛} = \sum_{𝑖 = 1}^{𝑛} 𝑎_{𝑖} 𝑏_{𝑖}

Example

$\vec{𝑎} = [\begin{matrix} 2 \\ - 1 \\ 3 \end{matrix}]$ , $\vec{𝑏} = [\begin{matrix} 4 \\ 5 \\ - 2 \end{matrix}]$ :

\vec{𝑎} \cdot \vec{𝑏} = 2 \cdot 4 + (- 1) \cdot 5 + 3 \cdot (- 2) = 8 - 5 - 6 = - 3

\vec{𝑎} \cdot \vec{𝑏} = ‖ \vec{𝑎} ‖ ‖ \vec{𝑏} ‖ cos 𝜃

where $𝜃$ is the angle between $\vec{𝑎}$ and $\vec{𝑏}$ — and $‖ \cdot ‖$ is the norm.

This links the algebraic dot product to geometric intuition: the dot product measures how much the vectors point in the same direction.

The square of the norm is the dot product of a vector with itself:

‖ \vec{𝑎} ‖^{2} = \vec{𝑎} \cdot \vec{𝑎}

See Norm for the full story.

Commutative:

\vec{𝑣} \cdot \vec{𝑤} = \vec{𝑤} \cdot \vec{𝑣}

Distributive over vector addition:

(\vec{𝑣} + \vec{𝑤}) \cdot \vec{𝑥} = \vec{𝑣} \cdot \vec{𝑥} + \vec{𝑤} \cdot \vec{𝑥}

Compatible with scalar multiplication:

(𝑐 \vec{𝑣}) \cdot \vec{𝑤} = 𝑐 (\vec{𝑣} \cdot \vec{𝑤})

Bilinear: linear in each argument separately (the three properties above combined).

For column vectors, the dot product is just matrix multiplication of ${\vec{𝑎}}^{𝑇}$ with $\vec{𝑏}$ :

\vec{𝑎} \cdot \vec{𝑏} = {\vec{𝑎}}^{𝑇} \vec{𝑏}

Norm — $‖ \vec{𝑣} ‖ = \sqrt{\vec{𝑣} \cdot \vec{𝑣}}$
Angles Between Vectors — $cos 𝜃 = \frac{\vec{𝑎} \cdot \vec{𝑏}}{‖ \vec{𝑎} ‖ ‖ \vec{𝑏} ‖}$
Cauchy–Schwarz Inequality
Triangle Inequality
Orthogonality — $\vec{𝑎} ⟂ \vec{𝑏} \Leftrightarrow \vec{𝑎} \cdot \vec{𝑏} = 0$
Cross Product + Dot vs Cross
Inner Product — generalization