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Question
If \[\vec{a,} \vec{b,} \vec{c}\] are three unit vectors such that \[\vec{a} \times \vec{b} = \vec{c} , \vec{b} \times \vec{c} = \vec{a,} \vec{c} \times \vec{a} = \vec{b} .\] Show that \[\vec{a,} \vec{b,} \vec{c}\] form an orthonormal right handed triad of unit vectors.
Solution
\[\text{ Given } :\]
\[ \vec{a} \times \vec{b} = \vec{c} \]
\[ \vec{b} \times \vec{c} = \vec{a} \]
\[ \vec{c} \times \vec{a} = \vec{b} . . . (1)\]
\[\text{ Now } ,\]
\[\left| \vec{a} \times \vec{b} \right| = \left| \vec{c} \right| = 1 (\because \vec{c} \text{ is a unit vector } )\]
\[\left| \vec{b} \times \vec{c} \right| = \left| \vec{a} \right| = 1 (\because \vec{a} \text{ is a unit vector } )\]
\[\left| \vec{c} \times \vec{a} \right| = \left| \vec{b} \right| = 1 (\because \vec{b} \text{ is a unit vector } )\]
\[ \therefore \left| \vec{a} \times \vec{b} \right| = \left| \vec{b} \times \vec{c} \right| = \left| \vec{c} \times \vec{a} \right| = 1 . . . (2)\]
\[ \text{ From (1) and (2), we know } \]
\[ \vec{a} , \vec{b} \text{ and } \vec{c} \text{ form an orthonormal right handed triad of unit vectors. } \]
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