[Math] Why does the dot product between two unit vectors equal the cosine on the angle between them

Question

I'm confused about why does the dot product between two unit vectors equal the cosine on the angle between them

Thank you.

Answer 1

Suppose the two unit vectors you have are given by $\langle \cos\theta,\sin\theta \rangle$ and $\langle \cos\phi,\sin\phi \rangle$. These are unit vectors because $\cos^2x+\sin^2x=1$ for any $x$. By definition, the angle between these two vectors would be $\theta-\phi$. Now, write the dot product: $$ \begin{split} \langle\cos\theta,\sin\theta\rangle\cdot\langle\cos\phi,\sin\phi\rangle &= \cos\theta\cos\phi+\sin\theta\sin\phi\\ &=^* \cos(\theta-\phi)\end{split}$$ Where $=^*$ follows because of the angle subtraction identity for cosine. Thus the dot product of two unit vectors is the cosine of the angle between them.