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Scalar triple product of vectors

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Question

$if \vec{a} and \vec{b} are prependicular unit vector and vector \vec{c} i s s u c h t h a t \vec{c} = \vec{a} + \vec{b},then the value of$ $(\vec{a} \times \vec{b}) . (\vec{b} \times \vec{c}) + (\vec{b} \times \vec{c}) . (\vec{c} \times \vec{a}) + (\vec{c} \times \vec{a}) . (\vec{a} \times \vec{b})$

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Solution

$(\vec{a} \times \vec{b}) . (\vec{b} \times \vec{c}) + (\vec{b} \times \vec{c}) . (\vec{c} \times \vec{a}) + (\vec{c} \times \vec{a}) . (\vec{a} \times \vec{b})$
$= (\vec{a} \cdot \vec{b}) (\vec{b} \cdot \vec{c}) - (\vec{a} \cdot \vec{c}) (\vec{b} \cdot \vec{b}) + (\vec{b} \cdot \vec{c}) (\vec{c} \cdot \vec{a}) - (\vec{b} \cdot \vec{a}) (\vec{c} \cdot \vec{c}) + (\vec{c} \cdot \vec{a}) (\vec{a} \cdot \vec{b}) - (\vec{c} \cdot \vec{b}) (\vec{a} \cdot \vec{a})$
= $0 - (\vec{a} . \vec{c}) + (\vec{b} . \vec{c}) (\vec{c} . \vec{a}) - 0 + 0 - (\vec{c} . \vec{b})$ $(\sin c e \vec{a} \cdot \vec{b} = 0 a n d |\vec{a}| = 1 a n d |\vec{b}| = 1)$
$= (\vec{b} . \vec{c}) (\vec{c} . \vec{a}) - (\vec{c} . \vec{a}) - (\vec{b} . \vec{c}) + 1 - 1$
$= ((\vec{b} . \vec{c}) - 1) ((\vec{c} . \vec{a}) - 1) - 1$ $(\sin c e \vec{b} \cdot \vec{c} = \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} = 0 + 1 = 1 a n d s i m i l a r l y \vec{c} \cdot \vec{a} = 1)$
$= (1 - 1) (1 - 1) - 1 = - 1$

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