Prove $\cos(\alpha − \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$ using vectors

Question

Suppose that $v$ and $w$ are unit vectors. If the angle between $v$ and $\hat{i}$ (the unit vector in the positive $x$ direction) is $\alpha$ and that between $w$ and $\hat{i}$ is $\beta$, prove that $$\cos(\alpha − \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$$

Answer 1

$v=\cos(\alpha)\hat{i}+\sin(\alpha)\hat{j}$ and $w=\cos(\beta)\hat{i}+\sin(\beta)\hat{j}$ . Now, the angle between $v$ and $w$ is $|\alpha-\beta|$. Thus, we have: $$v\cdot w=|v||w|\cos(\alpha-\beta)=\cos(\alpha-\beta)$$ $$(\cos(\alpha)\hat{i}+\sin(\alpha)\hat{j})\cdot (\cos(\beta)\hat{i}+\sin(\beta)\hat{j})=\cos(\alpha-\beta)$$ $$\cos(\alpha)\cos(\beta)+\sin(\alpha)\sin(\beta)=\cos(\alpha-\beta)$$