Inner Product


The inner product between two vectors $\vec a$ and $\vec b$ is denoted by $\vec a \cdot \vec b$, and defined as some action between the vectors that follows the rules:

  1. $\vec a \cdot \vec a \geqslant 0$
  2. $\vec a \cdot \vec a = 0$ if and only if $\vec a = 0$
  3. $\alpha \vec a\cdot\vec b = \vec a\cdot\alpha \vec b = \alpha \left( {\vec a\cdot\vec b} \right)$, where $\alpha \in \mathbb{R}$
  4. $\vec a\cdot\left( {\vec b + \vec c} \right) = \vec a\cdot\vec b + \vec a\cdot\vec c$, where $\vec c$ is vector.
  5. $\vec a\cdot\vec b = \vec b\cdot\vec a$


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