The **dot product** or **inner product** (or sometimes *scalar product*) of [[Vector|vectors]] $\boldsymbol v=\left(v_{1}, v_{2}\right)$ and $w=\left(w_{1}, w_{2}\right)$ is the number $\boldsymbol v \cdot \boldsymbol w$ : $ \boldsymbol v \cdot \boldsymbol w=v_{1} w_{1}+v_{2} w_{2} $ - [[commutative property|commutative]] because $\boldsymbol v\cdot \boldsymbol w=\boldsymbol w\cdot \boldsymbol v$ - $v_{1} w_{1}+v_{2} w_{2}=w_{1} v_{1}+w_{2} v_{2}$ because [[multiplication]] is commutative in a [[field]] - [[distributive property|distributive]] because for three vector $\boldsymbol a, \boldsymbol b, \boldsymbol c$, we know $\boldsymbol a\cdot(\boldsymbol b+\boldsymbol c)=\boldsymbol a\cdot\boldsymbol b+\boldsymbol a\cdot\boldsymbol c$ - Because multiplication is distributive in a field, and addition is commutative, the terms can be arranged to the desired result$\begin{aligned} (\boldsymbol b+\boldsymbol c)&=\left(b_{1}+c_{1}, \ldots, b_{n}+c_{n}\right)\\ \boldsymbol a \cdot( \boldsymbol b+ \boldsymbol c) &=a_{1}\left(b_{1}+c_{1}\right)+\ldots+a_{n}\left(b_{n}+c_{n}\right) \\ &=a_{1} b_{1}+a_{1} c_{1}+\ldots+a_{n} b_{n}+a_{n} c_{n} \\ &=a_{1} b_{1}+\cdots+a_{n} b_{n}+a_{1} c_{1}+\cdots+a_{n} c_{n}\\ &=\boldsymbol a\cdot\boldsymbol b+\boldsymbol a\cdot\boldsymbol c\end{aligned}$ - [[scalar multiplication]] can commute and associate freely through this product $x(\boldsymbol a \cdot \boldsymbol b)=(x\boldsymbol a)\cdot\boldsymbol b =\boldsymbol a\cdot x\boldsymbol b$ Sometimes instead of $\boldsymbol a \cdot \boldsymbol a$ we write $\boldsymbol a ^2$, even though a vector to any other power would not have any significance. When the vectors are in a [[vector space]] over the real numbers, $a^2\geq0$, but when over the [[Complex number|complex numbers]], $a^2$ might not be real. Useful for defining [[vector length]]