# Vector Identity and the Scalar Product

Ok, I have a question that probably has a very simple answer but for some reason I can't see it. Let $a$ and $r$ be two vectors of nonzero length with a common origin and let $\theta$ be the nonzero angle between them. Then, by definition of the cosine funtion, $$\cos \theta = \frac{|a|}{|r|}$$ where $|\cdot|$ denotes the norm. On the other hand, the scalar product is given by

$$\langle a, r \rangle = |a|\cdot |r| \cos \theta.$$

Putting these facts together we have $$\langle a, r \rangle = |a|\cdot |r| \cdot \frac{|a|}{|r|} = |a|\cdot |a| = |a|^2 = \langle a, a \rangle$$ which is a result that is independent of $r$ and thus makes no sense. What is my error?

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Your error is that $$\cos(\theta)=\frac{|a|}{|r|}$$ is not correct except when $a$ and $r$ are a leg and the hypotenuse of a right triangle, respectively, which is not the case in general.
Your definition for cosine is wrong. In $\mathbb{R}^2$, let $a = (0,1)$ and $r=(1,0)$ both radiating from the origin. According to your definition, this would give you $cos(\theta) = 1$ even though the angle between these two vectors is $\pi/2$ and $\cos(\pi/2) = 0$.