Dot product

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.

(“Dot Product” 2022)

Algebraic

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

(“Dot Product” 2022)

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

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