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  4. If veca ⋅ vecb =1, vecb ⋅ vecc =2 and vecc ⋅ veca =3, then the value of [ veca ×( vecb × vecc ), vecb ×( vecc × veca ), vecc ×( vecb × veca )] is :

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Q. If $\vec{a} \cdot \vec{b} =1, \vec{b} \cdot \vec{c} =2$ and $\vec{c} \cdot \vec{a} =3$, then the value of $[ \vec{a} \times( \vec{b} \times \vec{c} ), \vec{b} \times( \vec{c} \times \vec{a} ), \vec{c} \times( \vec{b} \times \vec{a} )]$ is :

JEE MainJEE Main 2022Vector Algebra

Solution:

$\vec{a} \times(\vec{b} \times \vec{c})=(\vec{a} \cdot \vec{c}) \vec{b}-(\vec{a} \cdot \vec{b}) \vec{c}=3 \vec{b}-\vec{c}$
$\vec{b} \times(\vec{c} \times \vec{a})=(\vec{b} \cdot \vec{a}) \vec{c}-(\vec{b} \cdot \vec{c}) \vec{a}=\vec{c}-2 \vec{a}$
$\vec{c} \times(\vec{b} \times \vec{a})=(\vec{c} \cdot \vec{a}) \vec{b}-(\vec{c} \cdot \vec{b}) \vec{a}=3 \vec{b}-2 \vec{a}$
${[3 \vec{b}-\vec{c}, \vec{c}-2 \vec{a}, 3 \vec{b}-2 \vec{a}] }$
$(3 \vec{b}-\vec{c}) \cdot[(\vec{c}-2 \vec{a}) \times(3 \vec{b}-2 \vec{a})]$
$(3 \vec{b}-\vec{c}) \cdot[3(\vec{c} \times \vec{b})-2(\vec{c} \times \vec{a})-6(\vec{a} \times \vec{b})]$
$-6[\vec{b}\,\, \vec{c}\,\, \vec{a}]+6[\vec{c}\,\, \vec{a}\,\, \vec{b}]$