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Vector triple product

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Question

$Let \vec{a}, \vec{b} and \vec{c} be three vectors such that | \vec{a} | = \sqrt{3}, |\vec{b}| = 5, \vec{b} \cdot \vec{c} = 10 and the angle between$
$\vec{b} and \vec{c} is \frac{π}{3} . If \vec{a} is perpendicular to the vector \vec{b} \times \vec{c}, then ∣ \vec{a} \times (\vec{b} \times \vec{c}) is equal to$

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Solution

$\begin{array}{l} | a | = \sqrt{3}, \bar{b} | = 5 \bar{b} . \bar{c} = 10 | \\ | \bar{b} | |c| \cos (\bar{b}, c) = 10 & 5. |c| |\cos \frac{π}{3} = 10| \bar{c} ∣ = 4 \\ (\bar{a}, \bar{b} \times \bar{c}) = 90 ° & | \bar{a} \times (\bar{b} \times \bar{c}) | = | \bar{a} | | \bar{b} \times c | \sin (\bar{a}, \bar{b} \times \bar{c}) \sqrt{3} . 5.4 \sin \frac{π}{3} \cdot \sin (\frac{π}{2}) \\ \sqrt{3} . 5 \cdot 4 \cdot \frac{\sqrt{3}}{2} = 30 \end{array}$

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