Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

If → a , → B , → C Are Mutually Perpendicular, Show that → C × ( → a × → B ) = 0 is the Converse True? - Physics

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If \[\vec{A} , \vec{B} , \vec{C}\] are mutually perpendicular, show that  \[\vec{C} \times \left( \vec{A} \times \vec{B} \right) = 0\] Is the converse true?

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Given: \[\vec{A} , \vec{B} \text{ and }\vec{C}\] are mutually perpendicular. \[\vec{A} \times \vec{B}\] is a vector with its direction perpendicular to the plane containing \[\vec{A} \text{ and } \vec{B}\]

∴ The angle between \[\vec{C} \text{ and } \vec{A} \times \vec{B}\] is either 0° or 180°.
i.e., \[\vec{C} \times \left( \vec{A} \times \vec{B} \right) = 0\] However, the converse is not true. For example, if two of the vectors are parallel, then also, \[\vec{C} \times \left( \vec{A} \times \vec{B} \right) = 0\] 

So, they need not be mutually perpendicular.

Concept: What is Physics?
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HC Verma Class 11, Class 12 Concepts of Physics Vol. 1
Chapter 2 Physics and Mathematics
Exercise | Q 16 | Page 29

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