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\vec{a} + \vec{b} = - \vec{c}

vector product is anti commutative:  \vec{a} \times \vec{b} = - \vec{b} \times \vec{a}

vector product is distributive over vector addition.
\vec{a} \times (\vec{b} +\vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}

vector  multiplied by itself is null vector.

so now LHS =
\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec {c} \times \vec{a} \\\\ \vec{a} \times \vec{b} + \vec{b} \times (-\vec{a} - \vec{b}) + (-\vec{a} - \vec{b}) \times \vec{a} \\\\ \vec{a} \times \vec{b} - \vec{b} \times \vec{a} - \vec{b} \times \vec{b} + -\vec{a} \times \vec{a} - \vec{b} \times \vec{a} \\\\ = 3 \vec{a} \times \vec{b} =3 \vec{b} \times \vec{c} =3\vec{c} \times \vec{a}

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