Let a, b and c be three vectors such that a = b ×( b × c ). If magnitudes of the vectors a, b and c are √ 2, 1 and 2 respectively and the angle…

Let $\displaystyle \vec{a}, \vec{b}$ and $\displaystyle \vec{c}$ be three vectors such that $\displaystyle \vec{a}=\vec{b} \times(\vec{b} \times \vec{c})$. If magnitudes of the vectors $\displaystyle \overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}$ and $\displaystyle \overrightarrow{\mathrm{c}}$ are $\displaystyle \sqrt{2}, 1$ and $\displaystyle 2$ respectively and the angle between $\displaystyle \overrightarrow{\mathrm{b}}$ and $\displaystyle \overrightarrow{\mathrm{c}}$ is $\displaystyle \theta\left(0<\theta<\frac{\pi}{2}\right)$, then the value of $\displaystyle 1+\tan \theta$ is equal to: