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Question
If \[\vec{c}\] is a unit vector perpendicular to the vectors \[\vec{a} \text{ and } \vec{b} ,\] write another unit vector perpendicular to \[\vec{a} \text{ and } \vec{b} .\]
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Solution
\[\vec{c} \text{ is a unit vector perpendicular to both } \vec{a} \text{ and } \vec{b} .\]
\[\Rightarrow \vec{c} =\frac{\vec{a} \times \vec{b}}{\left| \vec{a} \times \vec{b} \right|}\]
\[\Rightarrow- \vec{c} =\frac{\vec{b} \times \vec{a}}{\left| \vec{a} \times \vec{b} \right|}\]
\[\text{ Therefore,} - \vec{c} \text{ is perpendicular to } \vec{b} \text{ and } \vec{a .} \]
\[\text{ Thus } , - \vec{c} \text{ is another unit vector perpendicular to } \vec{a} \text{ and } \vec{b .}\]
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