Let $\vec{u}$, $\vec{v}$ and $\vec{w}$ be vectors in three-dimensional space, where $\vec{u}$ and $\vec{v}$ are unit vectors which are not perpendicular to each other and $\vec{u}\cdot\vec{w} = 1$, $\vec{v}\cdot\vec{w} = 1$, $\vec{w}\cdot\vec{w} = 4$. If the volume of the parallelepiped, whose adjacent sides are represented by the vectors $\vec{u}$, $\vec{v}$ and $\vec{w}$, is $\sqrt{2}$, then the value of $|3\vec{u} + 5\vec{v}|$ is _______.

$\text{Given:}\quad |\vec{u}| = 1,\quad |\vec{v}| = 1$

$\vec{u}\cdot\vec{v} \neq 0$

$\vec{u}\cdot\vec{w} = 1,\quad \vec{v}\cdot\vec{w} = 1,\quad \vec{w}\cdot\vec{w} = 4$

$\text{Volume} = \left|[\vec{u}\ \vec{v}\ \vec{w}]\right| = \sqrt{2}$

$\text{To find:}\quad \left|3\vec{u} + 5\vec{v}\right|$

$\text{Plan:}\quad (1)\ [\vec{u}\ \vec{v}\ \vec{w}]^2 = \det G = 2$ $(2)\ \text{find } p = \vec{u}\cdot\vec{v}$ $(3)\ \left|3\vec{u}+5\vec{v}\right|^2 = 34 + 30p$ $\text{Setup: the parallelepiped on } \vec{u},\ \vec{v},\ \vec{w}$ $\text{Volume} = \left|[\vec{u}\ \vec{v}\ \vec{w}]\right| = \sqrt{2}$ $[\vec{u}\ \vec{v}\ \vec{w}]^2 = \left(\sqrt{2}\right)^2 = 2$ $[\vec{u}\ \vec{v}\ \vec{w}]^2 = \det G$ $G = \begin{pmatrix} \vec{u}\cdot\vec{u} & \vec{u}\cdot\vec{v} & \vec{u}\cdot\vec{w} \\ \vec{v}\cdot\vec{u} & \vec{v}\cdot\vec{v} & \vec{v}\cdot\vec{w} \\ \vec{w}\cdot\vec{u} & \vec{w}\cdot\vec{v} & \vec{w}\cdot\vec{w} \end{pmatrix}$ $\text{Let } p = \vec{u}\cdot\vec{v}$ $G = \begin{pmatrix} 1 & p & 1 \\ p & 1 & 1 \\ 1 & 1 & 4 \end{pmatrix}$ $\det G = 1(4-1) - p(4p-1) + 1(p-1)$ $\det G = 3 - 4p^2 + p + p - 1$ $= 2 + 2p - 4p^2$ $\det G = [\vec{u}\ \vec{v}\ \vec{w}]^2 = 2$ $2 + 2p - 4p^2 = 2 \Rightarrow 2p - 4p^2 = 0$ $2p(1 - 2p) = 0 \Rightarrow p = 0 \text{ or } p = \dfrac{1}{2}$ $\vec{u}\cdot\vec{v} \neq 0 \Rightarrow p \neq 0$ $\therefore p = \dfrac{1}{2} \Rightarrow \cos\theta = \dfrac{1}{2} \Rightarrow \theta = 60^\circ$ $\left|3\vec{u}+5\vec{v}\right|^2 = (3\vec{u}+5\vec{v})\cdot(3\vec{u}+5\vec{v})$ $= 9|\vec{u}|^2 + 25|\vec{v}|^2 + 30\,\vec{u}\cdot\vec{v}$ $= 9(1) + 25(1) + 30\cdot\dfrac{1}{2}$ $= 34 + 15 = 49$ $\left|3\vec{u}+5\vec{v}\right| = \sqrt{49} = 7$