The dot product (also called the inner product) of two [[Vector Operations|Vectors]] is a scalar quantity obtained by multiplying corresponding components of the vectors and summing the results. For two vectors $\mathbf r$ and $\mathbf s$ where both $\in \mathbb R^n$: $ \mathbf r = \begin{bmatrix} r_1 \\ \vdots \\ r_n \end{bmatrix}, \quad \mathbf s= \begin{bmatrix} s_1 \\ \vdots \\ s_n \end{bmatrix} $ The dot product is calculated as: $ \mathbf r \cdot \mathbf s=(r_1*s_1)+\dots+ (r_n*s_n) $ **Dot product with itself:** When a vector is dotted with itself, the result is its squared [[Vector Length]]. ^8ea6e9 $ \mathbf r \cdot \mathbf r= r_1^2+\cdots +r_n^2 = \sqrt{r_1^2+\cdots +r_n^2}^2=\vert \vert \mathbf r \vert \vert ^2$ **Properties of the Dot Product:** - *Commutative:* The order of the vectors does not affect the result. $ \mathbf r \cdot \mathbf s = \mathbf s \cdot \mathbf r $ ^d17756 - *Distributive over addition:* The dot product distributes over vector addition. $ \mathbf r \cdot (\mathbf s+ \mathbf t) = \mathbf r \cdot \mathbf s+ \mathbf r \cdot \mathbf t $ ^065b11 - *Associative over scalar multiplication:* In general associative means, that we can group the variables (with parenthesis) in any kind of way. Here $a$ is a scalar, which can be pulled out, and gets multiplied by the remaining dot product. $ (a \mathbf r) \cdot \mathbf s = a*(\mathbf r\cdot \mathbf s) $