In mathematics, the Dot product or Scalar product is an algebraic operation that takes two equal-length sequences of numbers (usually Coordinate vector), and returns a single number.

## Algebraic

Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers.

\(\vec{a} \cdot \vec{b} = \sum^{n}_{i=1} \vec{a}_i\vec{b}_i\)

## Geometric

The dot product of two vectors defines the lengths of, and angle between, the two vectors.

\(\vec{a} \cdot \vec{b} = \|a\|\|b\|\href{/posts/cosine}{\cos} \theta\)

- Vector length: \(\lVert\vec{x}\rVert\) \(=\) \(\sqrt{\vec{x} \cdot \vec{x}}\)
- Angles (where \(\theta\) is the angle between \(\vec{x}\) and \(\vec{y}\)): \(\href{/posts/cosine}{\cos}(\theta) = \frac{\vec{x} \cdot \vec{y}}{\lVert\vec{x}\rVert\lVert\vec{y}\rVert}\)

## Bibliography

“Dot Product.” 2022.

*Wikipedia*, July. https://en.wikipedia.org/w/index.php?title=Dot_product&oldid=1096253368.