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If two vectors $a$ and $b$ make angle $\alpha$ and $\beta$ with the $x$-axis, prove, using vectors, that: $$\cos(\beta - \alpha) = \cos \alpha \cos\beta + \sin\alpha \sin\beta$$

I tried taking components of each vector along the plane, but that didn't get me anywhere. I tried taking component of vector $b$ along vector $a$, but I also got stuck there. Can anyone point me in the right direction?

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Let $u = \langle\cos \alpha,\sin \alpha\rangle$ and $v = \langle\cos \beta,\sin \beta\rangle$.

Then, $\cos \theta = \dfrac{u \cdot v}{\|u\|\|v\|}$ where $\theta$ is the angle between the two vectors $u$ and $v$.

Now, plug in $u$, $v$, and $\theta$ in terms of $\alpha$ and $\beta$ and see what you get.

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