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Question
Show that the vectors \[\vec{a,} \vec{b,} \vec{c}\] are coplanar if and only if \[\vec{a} + \vec{b}\], \[\vec{b} + \vec{c}\] and \[\vec{c} + \vec{a}\] are coplanar.
Solution
Consider,
\[\left[ \left( \vec{a} + \vec{b} \right) \left( \vec{b} + \vec{c} \right) \left( \vec{c} + \vec{a} \right) \right] = \left( \vec{a} + \vec{b} \right) . \left\{ \left( \vec{b} + \vec{c} \right) \times \left( \vec{c} + \vec{a} \right) \right\} \]
\[ = \left( \vec{a} + \vec{b} \right) . \left( \vec{b} \times \vec{c} + \vec{b} \times \vec{a} + \vec{c} \times \vec{c} + \vec{c} + \vec{a} \right)\]
\[ = \left( \vec{a} + \vec{b} \right) . \left( \vec{b} \times \vec{c} + \vec{b} \times \vec{a} + \vec{c} \times \vec{a} \right) ( \because \vec{c} \times \vec{c} = \vec{0} )\]
\[ = \vec{a} . \left( \vec{b} \times \vec{c} \right) + \vec{a} . \left( \vec{b} \times \vec{a} \right) + \vec{a} . \left( \vec{c} \times \vec{a} \right) + \vec{b} . \left( \vec{b} \times \vec{c} \right) + \vec{b} . \left( \vec{b} \times \vec{a} \right) + \vec{b} . \left( \vec{c} \times \vec{a} \right)\]
\[ = \vec{a} . \left( \vec{b} \times \vec{c} \right) + \vec{b} . \left( \vec{c} \times \vec{a} \right)\]
\[ = \left[ \vec{a} \vec{b} \vec{c} \right] + \left[ \vec{b} \vec{c} \vec{a} \right]\]
\[ = 2\left[ \vec{a} \vec{b} \vec{c} \right]\]
Now, we can see that
\[\left[ ( \vec{a} + \vec{b} ) ( \vec{b} + \vec{c} ) ( \vec{c} + \vec{a} ) \right] = 2\left[ \vec{a} \vec{b} \vec{c} \right]\]
Hence, the vectors
\[\vec{a} , \vec{b} , \vec{c}\] are coplanar if and only if \[\vec{a} + \vec{b} , \vec{b} + \vec{c} , \vec{c} + \vec{a}\]are coplanar.
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