$\text{Let }\stackrel{\rightharpoonup }{u} be a vector on rectangular coordinate system with sloping angle {60}^0, suppose that \left|\overrightarrow{u}-\hat{i}\right| is geometric mean of$

$\left|\overline{u}\right| and \left|\overline{u}-2\hat{i}\right| where \hat{i} is the unit vector along x-axis then the value of (\sqrt{2}+1) \left|\overline{u}\right| is$

$$\begin{gathered} \text{∵angle between }\overrightarrow{u}and \hat{i} is {60}^0 \\ \overrightarrow{ u} . \hat{i} =\left|\overrightarrow{u}\right|\left|\hat{i}\right| cos {60}^0=\frac{\left|\overrightarrow{u}\right|}{2} \\ given that \left|\overrightarrow{u}\right|, \left|\overrightarrow{u}-\hat{i}\right|, \left|\overrightarrow{u}-2\hat{i}\right| are in G.P. \\ {\left|\overrightarrow{u}-\hat{i}\right|}^2 = \left|\overrightarrow{u}\right| \left|\overrightarrow{u}-2\hat{i}\right| \\ squarring on both sides \end{gathered}$$

$\begin{array}{l}\text{}{\left[{\left|\overrightarrow{u}\right|}^2+ {\left|\widehat{i}\right|}^2-2 \left|\overline{u}.\widehat{i}\right|\right]}^2={\left|\overrightarrow{u}\right|}^2 \left[{\left|\overrightarrow{u}\right|}^2+4{\left|\widehat{i}\right|}^2-4 \left|\overrightarrow{u}.\widehat{i}\right|\right]\\ \text{}{\left[{\left|\overrightarrow{u}\right|}^2+ 1-2 \frac{\left|\overrightarrow{u}\right|}{2}\right]}^2={\left|\overrightarrow{u}\right|}^2 \left[{\left|\overrightarrow{u}\right|}^2+4x1-4\frac{\left|\overrightarrow{u}\right|}{2}\right]\\ \text{}{\left|\overrightarrow{u}\right|}^2+2\left|\overrightarrow{u}\right|-1=0\\ \text{}\left|\overrightarrow{u}\right| =\frac{-2\pm 2\sqrt{2}}{2}\\ \text{}\left|\overrightarrow{u}\right|=\sqrt{2}-1\end{array}$