Note that $\vec{a} \cdot \hat{b} = \left \| a \right \| \left ( \hat{a} \cdot \hat{b}\right ) \;\;and \;\;\; \hat{a} \cdot \hat{b}$ is dot product of unit vectors, which means that
It is component of one unit vector along the other, which will always be between -1 and 1 .
Hence,
$ \;\;\;\;-1 \leqslant \hat{a} \cdot \hat{b} \leqslant 1$
$\Rightarrow -\left \| \vec{a} \right \| \leqslant \vec{a} \cdot \hat{b} \leqslant \left \| \vec{a} \right \|$
$\Rightarrow -\left \| a \right \| \leqslant \left \langle a,b \right \rangle \leqslant \left \| a \right \|$ (In our question)
Rest all of the other options can't be implied with the limited information provided.
What about $ \left \langle a,b \right \rangle \leqslant \left \| b \right \|$ ?
It is obviously incorrect as $\left \| b \right \|$ is 1, and we don't know $\left \| a \right \|$, to be able to comment on $ \left \langle a,b \right \rangle \leqslant \left \| b \right \|$
So, answer is option A