Let $\overrightarrow a$, $\overrightarrow b$ and $\overrightarrow c$ be three vectors such that $\left| {\overrightarrow a } \right| = \sqrt 3$, $$\left| {\overrightarrow b } \right| = 5,\overrightarrow b .\overrightarrow c = 10$$ and the angle between $\overrightarrow b$ and $\overrightarrow c$ is ${\pi \over 3}$. If ${\overrightarrow a }$ is perpendicular to the vector $\overrightarrow b \times \overrightarrow c$ , then $$\left| {\overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right)} \right|$$ is equal to _____.

Given $\left| {\overrightarrow a } \right| = \sqrt 3$, $\left| {\overrightarrow b } \right| = 5$ <br><br>Given $\overrightarrow b .\overrightarrow c = 10$ <br><br>And the angle between $\overrightarrow b$ and $\overrightarrow c$ is ${\pi \over 3}$ <br><br>$\therefore$ $bc\cos {\pi \over 3}$ = 10 <br><br>$\Rightarrow$ c = 4 <br><br>${\overrightarrow a }$ is perpendicular to the vector $\overrightarrow b \times \overrightarrow c$ <br><br>$\therefore$ $\overrightarrow a .\left( {\overrightarrow b \times \overrightarrow c } \right)$ = 0 and angle between them is ${\pi \over 2}$ <br><br>Now $$\left| {\overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right)} \right|$$ <br><br>= $$\left| {\overrightarrow a } \right|\left| {\overrightarrow b \times \overrightarrow c } \right|\sin {\pi \over 2}$$ <br><br>= $\left| {\overrightarrow a } \right|$.${\left| {\overrightarrow b } \right|.\left| {\overrightarrow c } \right|}$$\sin {\pi \over 3}$.1 <br><br>= $\sqrt 3 \times 5 \times 4 \times {{\sqrt 3 } \over 2}$ <br><br>= 30