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Let $a , b \in \mathbb{R}^n$ and let $\alpha$ be the angle formed between $a$ and $b$. Is it true that:

$$ \| a \| \| b \| \cos \alpha = \langle a,b\rangle $$

($\langle\cdot,\cdot\rangle$ being the dot product)

If so why?

kennytm
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aribaldi
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    This is defined / proved / discussed in virtually any elementary text on linear algebra that introduces the dot product, as well as the Wikipedia article on the dot product. Have you done any research on this question, and what did you find? –  May 14 '16 at 06:32

1 Answers1

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That's taken as the definition of $\cos \alpha$ in order to define the angle between two vectors in $\mathbb R^n$. Note that it is well defined since $|<a,b>| \le ||a|| ||b||$ by C-S inequality.

Shubhodip Mondal
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