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Q. If $\vec{a}, \vec{b}, \vec{ c }$ are three non-zero vectors and $\hat{n}$ is a unit vector perpendicular to $\vec{ c }$ such that $\vec{ a }=\alpha \vec{ b }-\hat{ n },(\alpha \neq 0) $ and $ \vec{ b } \cdot \vec{ c }=12$, then $|\vec{ c } \times(\vec{ a } \times \vec{ b })|$ is equal to :

JEE MainJEE Main 2023Vector Algebra

Solution:

$ \hat{ n } \perp \vec{ c } \,\,\, \vec{ a }=\alpha \vec{ b }-\vec{ n } $
$ \vec{b} \cdot \vec{c}=12$
$ \vec{ a } \cdot \vec{ c }=\alpha(\vec{ b } \cdot \vec{ c })-\vec{ n } \cdot \vec{ c } $
$ \vec{a} \cdot \vec{c}=\alpha(\vec{b} \cdot \vec{c}) $
$ |\vec{c} \times(\vec{a} \times \vec{b})|=|(\vec{c} \cdot \vec{b}) \vec{a}-(\vec{c} \cdot \vec{a}) \vec{b}|$
$ =|(\vec{ c } \cdot \vec{ b }) \vec{ a }-\alpha(\vec{ b } \cdot \vec{ c }) \vec{ b }| $
$ =|(\vec{c} \cdot \vec{b})||\vec{ a }-\alpha \vec{ b }| $
$ =12 \times(|\vec{ n }|) $
$ =12 \times 1 $
$ =12 $