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?
Best Answer
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 $$ \begin{align*} |\textbf{a}- \textbf{b}|^2 &= (\textbf{a}- \textbf{b}) \cdot (\textbf{a}- \textbf{b})\\ &= \textbf{a} \cdot \textbf{a} - 2 (\textbf{a} \cdot \textbf{b}) + \textbf{b} \cdot \textbf{b} \\ &= |\textbf{a}|^2 - 2 (\textbf{a} \cdot \textbf{b}) + |\textbf{b}|^2 \end{align*} $$
Simplify this equation, and you will get the desired formula.