Dot product

Definition

Let $n \ge 1$ and let $x = (x_1, \dots, x_n)$ and $y = (y_1, \dots, y_n)$ be vectors of $\R^n$. The dot product of $x$ and $y$ is the real number $$x \cdot y = \sum_{i=1}^{n} x_i y_i = x_1 y_1 + x_2 y_2 + \cdots + x_n y_n.$$

Remarks

In matrix notation, $x \cdot y = x^T y$ for column vectors $x$ and $y$. It is symmetric, bilinear and positive definite, so it is an inner product on $\R^n$. Geometrically, $x \cdot y = \norm{x} \norm{y} \cos\theta$, where $\theta$ is the angle between $x$ and $y$. The same formula defines a symmetric bilinear form on $F^n$ for every field $F$.

Explanations

L1 human

The dot product multiplies two vectors coordinate by coordinate and adds the results. For example, $(1, 2, 3) \cdot (4, 5, 6) = 4 + 10 + 18 = 32$. It is zero exactly when the two vectors are perpendicular, or when one of them is zero.