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  4. Let veca, vecb and vecc be three non zero vectors such that vecb ⋅ vecc=0 and veca ×( vecb × vecc)=( vecb- vecc/2). If vecd be a vector such that vecb ⋅ vecd= veca ⋅ vecb, then ( veca × vecb) ⋅( vecc × vecd) is equal to

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Q. Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non zero vectors such that $\vec{b} \cdot \vec{c}=0$ and $\vec{a} \times(\vec{b} \times \vec{c})=\frac{\vec{b}-\vec{c}}{2}$. If $\vec{d}$ be a vector such that $\vec{b} \cdot \vec{d}=\vec{a} \cdot \vec{b}$, then $(\vec{a} \times \vec{b}) \cdot(\vec{c} \times \vec{d})$ is equal to

JEE MainJEE Main 2023Vector Algebra

Solution:

$ (\vec{a} \cdot \vec{c}) \vec{b}-(\vec{a} \cdot \vec{b}) \vec{ c }=\frac{\vec{b}-\vec{c}}{2} $
$ \vec{a} \cdot \vec{c}=\frac{1}{2}, \vec{a} \cdot \vec{b}=\frac{1}{2}$
$\therefore \vec{b} \cdot \vec{d}=\frac{1}{2}$
$ (\vec{a} \times \vec{b}) \cdot(\vec{c} \times \vec{d})=\vec{a} \cdot(\vec{b} \times(\vec{c} \times \vec{d})) $
$ =\vec{a} \cdot((\vec{b} \cdot \vec{d}) \vec{c}-(\vec{b} \cdot \vec{c}) \vec{d})$
$ =(\vec{a} \cdot \vec{c})(\vec{b} \cdot \vec{d})=\frac{1}{4}$