Dot Product · stemformulas

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For two vectors $\small \vec{a}$ and $\small \vec{b}$ in $\small \mathbb{R}^n$ (real coordinate spaces, the dot product is defined as

$$ \vec{a} \cdot \vec{b} = \sum_{i=1}^n a_i b_i $$

Where

$\small a_i$ and $\small b_i$ are the $\small i$-th components of $\small \vec{a}$ and $\small \vec{b}$, respectively, and
$\small n$ is the dimension of $\small \vec{a}$ and $\small \vec{b}$.

Alternative definition #

Given that $\small \vec{a}$ and $\small \vec{b}$ are vectors with lengths $\small |a|$ and $\small |b|$, respectively, the dot product is also equal to

$$ \vec{a} \cdot \vec{b} = |a| |b| \cos \theta $$

Where $\small \theta$ is the angle between $\small \vec{a}$ and $\small \vec{b}$.

Note that the dot product is a less general version of the inner product.

Properties #

$\small \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$
$\small (\vec{a} + \vec{b}) \cdot \vec{c} = \vec{a} \cdot \vec{c} + \vec{b} \cdot \vec{c}$
$\small \vec{a} \cdot \vec{a} = |a|^2$
$\small \vec{a} \cdot \vec{b} = 0$ if and only if $\small \vec{a}$ and $\small \vec{b}$ are orthogonal
For a scalar s, $(\small s \vec{a}) \cdot \vec{b} = s \cdot (\vec{a} \cdot \vec{b})$

Alternative definition #

Properties #

Sources #