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I am having a problem with an exercise. In a certain euclidean space, the angle between two vectors $u$ and $v$ is said to be $\theta$ which satisfies the equation:

$$\cos\theta=\frac{(u,v)}{\|u\|\|v\|}$$

Why is the angle a well-defined quantity if $\theta \in [0;\pi]$?

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3  
It isn't, unless you also add the condition $0\le\theta\le\pi$. –  Gerry Myerson Feb 10 '13 at 1:55
    
Yes we have this condition –  Carpediem Feb 10 '13 at 1:55
    
@user43758: What does the graph of $\cos$ look like? –  wj32 Feb 10 '13 at 1:58
    
If theta is between 0 and pi, cos is a decreasing function –  Carpediem Feb 10 '13 at 2:00
    
Can someone help me? –  Carpediem Feb 10 '13 at 2:03

1 Answer 1

up vote 3 down vote accepted

It's not.

But suppose you add the restriction that $\|u\| >0, \|v\|>0$. By the Cauchy-Schwarz inequality, $|(u,v)| \leq \|u\|\|v\|$, so $$-1 \leq \frac{(u,v)}{\|u\|\|v\|} \leq 1.$$

It follows that $\theta\in [0,\pi]$ exists and is unique, since $\cos$ is a bijection from $[0,\pi]$ to $[-1,1]$.

$\hspace{3cm}$cos(x) is a bijection

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theta is between 0 and $\pi$ –  Carpediem Feb 10 '13 at 2:05
    
Yes. Are you still confused about a part of the answer? –  user7530 Feb 10 '13 at 2:07
    
Yes. In the question they already told be the interval between which theta is defined. Then they are asking me why is the angle a well defined quantity? –  Carpediem Feb 10 '13 at 2:09
1  
@user43758: It's well-defined because $\cos:[0,\pi]\to[-1,1]$ is a bijection (as stated in this answer). –  wj32 Feb 10 '13 at 2:12
1  
@user7530: I hope I didn't modify too much. –  robjohn Feb 10 '13 at 11:45

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