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Question
For non-zero vectors \[\vec{a,} \vec{b} \text { and }\vec{c}\] the relation \[\left| \left( \vec{a} \times \vec{b} \right) \cdot \vec{c} \right| = \left| \vec{a} \right| \left| \vec{b} \right| \left| \vec{c} \right|\] holds good, if
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\[\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{c} = 0\]
\[\vec{a} \cdot \vec{b} = 0 = \vec{c} \cdot \vec{a}\]
\[\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{c} = \vec{c} \cdot \vec{a} = 0\]
\[\vec{b} \cdot \vec{c} = \vec{c} \cdot \vec{a} = 0\]
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Solution
\[ \vec{a} . \vec{b} = \vec{b} . \vec{c} = \vec{c} . \vec{a} = 0\]
We have
\[\left| \left( \vec{a} \times \vec{b} \right) . \vec{c} \right| \]
\[ = \left| \left( \vec{a} \times \vec{b} \right) \right| \left| \vec{c} \right|\left| cos\theta \right|\]
\[ = \left| \left( \vec{a} \times \vec{b} \right) \right| \left| \vec{c} \right| \left( \text { If } \theta = 0^\circ \text { or } 180^\circ , \text { i . e . vectors } \vec{a} \times \vec{b} \text { and }\vec {c}\text { are parallel } \right)\]
\[ = \left| \left( \left| \vec{a} \right|\left| \vec{b} \right| \sin \alpha \right) \right|\left| \vec{c} \right|\]
\[ = \left| \vec{a} \right|\left| \vec{b} \right|\left| \vec{c} \right| \left( \text { If } \alpha = 90^\circ,\text { i . e . vectors }\vec{a}\text { and } \vec{b} \text { are perpendicular } \right)\]
\[ \therefore \left| \left( \vec{a} \times \vec{b} \right) . \vec{c} \right| = \left| \vec{a} \right|\left| \vec{b} \right|\left| \vec{c} \right| \left(\text { If vectors } \vec{a} , \vec{b} , \vec{c} \text { are perpendicular to each other } \right)\]
\[\text { Thus, the relation } \left| \left( \vec{a} \times \vec{b} \right) . \vec{c} \right| = \left| \vec{a} \right|\left| \vec{b} \right|\left| \vec{c} \right|\text { holds good if } \vec{a} . \vec{b} = 0 , \vec{b} . \vec{c} = 0 \text { and } \vec{c} . \vec{a} = 0 .\]
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