If $\vec{a}, \vec{b} and \vec{c}$ are the position vectors of the vertices A, B and C respectively of a $∆ A B C such that |\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}| = 20,$ then area of $∆ A B C$ is ____________.

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Solution

$For \vec{a}, \vec{b} and \vec{c} position vectors of A, B and C vertices of a Δ A B C, given |\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}| = 20 Area of Δ = \frac{1}{2} |\vec{A B} \times \vec{A C}| Since \vec{A B} = \vec{b} - \vec{a} and \vec{A C} = \vec{c} - \vec{a} i . e . \vec{A B} \times \vec{A C} = (\vec{b} - \vec{a}) \times (\vec{c} - \vec{a}) = \vec{b} \times \vec{c} - \vec{b} \times \vec{a} - \vec{a} \times \vec{c} + \vec{a} \times \vec{a} = \vec{b} \times \vec{c} + \vec{a} \times \vec{b} + \vec{c} \times \vec{a} + 0 \{\begin{cases} ∴ \vec{a} \times \vec{b} = - \vec{b} \times \vec{a} \\ \vec{a} \times \vec{c} = - \vec{c} \times \vec{a} \\ \vec{a} \times \vec{a} = 0 \end{cases} = \vec{b} \times \vec{c} + \vec{a} \times \vec{b} + \vec{c} \times \vec{a} i . e . \frac{1}{2} |\vec{A B} \times \vec{A C}| = \frac{1}{2} |\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}| = \frac{1}{2} \times 20 (given) i . e . Area of Δ A B C = 10$

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