Say I have $\vec a$ a nonzero vector. After calculations for a question, I have that, for any $\vec u$,$\vec v$,
$$\vec u \cdot \vec v=\lvert\vec a\rvert^2(\vec u\cdot\vec v)+\beta(\vec a \cdot \vec u)+\alpha(\vec a \cdot \vec v)$$
Can I then effectively equate coefficients and say $\lvert a\rvert=1, \alpha=\beta=0$?