Definition:Dot Product/Real Euclidean Space

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Let $\mathbf a$ and $\mathbf b$ be vectors in real Euclidean space $\R^n$.

The dot product of $\mathbf a$ and $\mathbf b$ is defined as:

$\mathbf a \cdot \mathbf b = \norm {\mathbf a} \, \norm {\mathbf b} \cos \angle \mathbf a, \mathbf b$


$\norm {\mathbf a}$ denotes the length of $\mathbf a$
$\angle \mathbf a, \mathbf b$ is the angle between $\mathbf a$ and $\mathbf b$, taken to be between $0$ and $\pi$.

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