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Vector Cosine

The Cauchy-Schwarz Inequality can be written

$\displaystyle \frac{\left\vert\left<\underline{u},\underline{v}\right>\right\vert}{\Vert\underline{u}\Vert\cdot\Vert\underline{v}\Vert} \leq 1. $

In the case of real vectors $ \underline{u},\underline{v}$, we can always find a real number $ \theta\in[0,\pi]$ which satisfies

$\displaystyle \zbox {\cos(\theta) \isdef \frac{\left<\underline{u},\underline{v}\right>}{\Vert\underline{u}\Vert\cdot\Vert\underline{v}\Vert}.} $

We thus interpret $ \theta$ as the angle between two vectors in $ {\bf R}^N$.

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Triangle Difference Inequality