We know that " The resultant of two equal vectors bisects the angle between them ".
In the given question, $$\overrightarrow{w} = \overrightarrow{u} + \alpha \overrightarrow{v} $$
Here, $\overrightarrow{w}$ is the resultant of two vectors $\overrightarrow{u}$ and $\alpha$$\overrightarrow{v}$.
Thus, for $\overrightarrow{w}$ to bisect the angle between $\overrightarrow{u}$ and $\overrightarrow{v}$, magnitude of the two vectors must be equal.
$\left \| u \right \| =\left \| \alpha v \right \|$ $\Rightarrow$ $2\left \| v \right \| =\alpha \left \| v \right \|$
$\therefore$ $\alpha = 2$