Explanation

(c)
Since, $\left| \overrightarrow{\mathbf{u}} \right| = 1,\ \left| \overrightarrow{\mathbf{v}} \right| = 2,\ \left| \overrightarrow{\mathbf{w}} \right| = 3$
The projection of $\overrightarrow{\mathbf{v}}$ along $\overrightarrow{\mathbf{u}} = \frac{\overrightarrow{\mathbf{v}} \bullet \overrightarrow{\mathbf{u}}}{\left| \overrightarrow{\mathbf{u}} \right|}$
and the projection of $\overrightarrow{\mathbf{w}}\ along\ \overrightarrow{\mathbf{u}} = \frac{\overrightarrow{\mathbf{w}} \bullet \overrightarrow{\mathbf{u}}}{\left| \overrightarrow{\mathbf{u}} \right|}$
according to given condition,
$\frac{\overrightarrow{\mathbf{v}} \bullet \overrightarrow{\mathbf{u}}}{\left| \overrightarrow{\mathbf{u}} \right|} = \frac{\overrightarrow{\mathbf{w}} \bullet \overrightarrow{\mathbf{u}}}{\left| \overrightarrow{\mathbf{u}} \right|} \Longrightarrow \overrightarrow{\mathbf{v}} \bullet \overrightarrow{\mathbf{u}} = \overrightarrow{\mathbf{w}} \bullet \overrightarrow{\mathbf{u}}\ \ \ \ \ldots.(i)$
Also, $\overrightarrow{\mathbf{v}} \bullet \overrightarrow{\mathbf{w}} = 0$
Now, |$\overrightarrow{\mathbf{u}} - \overrightarrow{\mathbf{v}} + \overrightarrow{\mathbf{w}}|^{2} = |{\overrightarrow{\mathbf{u}}|}^{2} + \left| \overrightarrow{\mathbf{v}} \right|^{2} + \left| \overrightarrow{\mathbf{w}} \right|^{2}$
$- 2\overrightarrow{\mathbf{u}} \bullet \overrightarrow{\mathbf{v}} - 2\overrightarrow{\mathbf{v}} \bullet \overrightarrow{\mathbf{w}} + 2\overrightarrow{\mathbf{u}} \bullet \overrightarrow{\mathbf{w}}$
$= 1 + 4 + 9 - 2\overrightarrow{\mathbf{u}} \bullet \overrightarrow{\mathbf{v}} + \overrightarrow{\mathbf{v}} \bullet \overrightarrow{\mathbf{u}}\ \ \ \ \$[from Eq. (i)
$\Longrightarrow |\overrightarrow{\mathbf{u}} - \overrightarrow{\mathbf{v}} + \overrightarrow{\mathbf{w}}|^{2} = 14 + 0$
$\Longrightarrow \left| \overrightarrow{\mathbf{u}} - \overrightarrow{\mathbf{v}} + \overrightarrow{\mathbf{w}} \right| = \sqrt{14}$