Let \( \overrightarrow{\mathrm{u}}, \overrightarrow{\mathrm{v}} \) ...

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Let \( \overrightarrow{\mathrm{u}}, \overrightarrow{\mathrm{v}} \) and \( \overrightarrow{\mathrm{w}} \) be vectors in three dimensional
\( \mathrm{P} \) space, where \( \vec{u} \) and \( \vec{v} \) are unit vectors which are not

W perpendicular to each other and \( \overrightarrow{\mathrm{u}} \cdot \overrightarrow{\mathrm{W}}=1, \overrightarrow{\mathrm{v}} \cdot \overrightarrow{\mathrm{W}}=1 \), \( \overrightarrow{\mathrm{W}} . \overrightarrow{\mathrm{W}}=4 \). If the volume of the parallelepiped, whose adjacent sides are represented by the vectors \( \vec{u} \vec{v} \) and \( \overrightarrow{\mathrm{W}} \) is \( \sqrt{2} \), then the value of \( |3 \vec{u}+5 \overrightarrow{\mathrm{v}}| \) is
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