For two vectors $a$ and $b$, why does $\cos(θ)$ equal the dot product of $a$ and $b$ divided by the product of the vectors' magnitudes? While watching a video about dot products (https://www.youtube.com/watch?v=WDdR5s0C4cY), the following formula is presented for finding the angle between two vectors:


  For vectors $a$, and $b$,
  $$\cos( \theta ) =  \frac{(a, b)}{  \|  a  \|   \| b \| }$$
  where $(a,b)$ is the dot product of $a$ and $b$.


How is this formula derived?
There are several derivations of this online. Here's where you can start. 

Define two vectors $\textbf{a}$ and $\textbf{b}$. Then $ \textbf{a} - \textbf{b}$ is the vector that connects their endpoints and makes a triangle. 

Therefore, we have a triangle with side lengths  $|\textbf{a}|$, $|\textbf{b}|$, and $|\textbf{a} - \textbf{b}|$. Let the angle between the two vectors be $\theta$. By the Law of Cosines, we have

$$|\textbf{a} - \textbf{b}|^2 = |\textbf{a}|^2 + |\textbf{b}|^2 - 2 |\textbf{a}|  |\textbf{b}| \cos (\theta)$$

Now, use the fact that 
$$
\begin{align*}
|\textbf{a}- \textbf{b}|^2 &amp;= (\textbf{a}- \textbf{b}) \cdot (\textbf{a}- \textbf{b})\\ &amp;= \textbf{a} \cdot \textbf{a} - 2 (\textbf{a} \cdot   \textbf{b}) + \textbf{b} \cdot \textbf{b} \\ &amp;=  |\textbf{a}|^2 - 2 (\textbf{a} \cdot   \textbf{b}) + |\textbf{b}|^2
\end{align*}
$$

Simplify this equation, and you will get the desired formula.
Dot product of two vectors is defined as the product of the magnitude of the two vectors together with the cosine of the angle between the two vectors. Mathematically,
a.b=|a||b|cos§