Dot product
High school
Consider two vectors $x = (x_1, \dots, x_n)$ and $y = (y_1, \dots, y_n)$ in $\reals^n$. The dot product of $x$ and $y$ is
\[x \cdot y = x_1 y_1 + x_2 y_2 + \dots + x_n y_n.\]
Undergraduate
Consider the vector space $\reals^n$, with vectors written as columns. The dot product is the standard inner product on $\reals^n$,
\[x \cdot y = x^{\top} y.\]Examples
Examples
-
$(x, y) \mapsto x_1 y_1 + x_2 y_2$ on $\reals^2$
Undergraduate
$(x, y) \mapsto x_1 y_1 + x_2 y_2$ is the dot product on $\reals^2$.
Non-examples
Things that may be mistaken for examples.
-
$(x, y) \mapsto 2 x_1 y_1 + x_2 y_2$ on $\reals^2$
Undergraduate
$(x, y) \mapsto 2 x_1 y_1 + x_2 y_2$ is not the dot product, because of the factor $2$.
Properties
- $x \cdot y = \norm{x} \norm{y} \cos \theta$, where $\theta$ is the angle between $x$ and $y$. In particular, $x \cdot y = 0$ exactly when $x$ and $y$ are perpendicular (or one of them is zero).
Distinctions involving this term
- Dot product vs. inner product vs. bilinear form: Bilinear forms, inner products, and the dot product are all ways of multiplying two vectors to get a number, from the most general to the most specific. The dot product is one particular inner product on $\reals^n$. An inner product is a bilinear form that is also symmetric and positive definite.