Mathematics

Problem

Let

\mathbf{u}

be a vector on rectangular coordinate system with sloping angle

60^\circ

. Suppose that

|\mathbf{u} - \hat{i}|

is geometric mean of

|\mathbf{u}|

and

|\mathbf{u} - 2\hat{i}|

, where

\hat{i}

is the unit vector along the

X

-axis. Then, the value of

(\sqrt{2} + 1)|\mathbf{u}|