Let $u$ and $v$ be two vectors in $\mathbf{R}^2$ whose Euclidean norms satisfy $\|u\| = 2\|v\|$. What is the value of $\alpha$ such that $w = u + \alpha v$ bisects the angle between $u$ and $v$?

A vector that bisects the angle between two non-zero vectors $u$ and $v$ lies along the sum of their unit vectors:

Multiplying by $\|u\|$:

Comparing this with $w = u + \alpha v$, we have:

Given that $\|u\| = 2\|v\|$, we find:

Hence, the correct option is (A).