Q55 / 120 · VITEEE 2017 · Mathematics

If $((\vec{a} \times \vec{b}) \times(\vec{c} \times \vec{d})) \cdot(\vec{a} \times \vec{d})=0$, then which of the following is always true?

  • A
    $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ are necessarily coplanar
  • B
    either $\vec{a}$ or $\vec{d}$ must lie in the plane of $\vec{b}$ and $\vec{c}$
  • Correct answer
    C
    either $\vec{b}$ or $\vec{c}$ must lie in the plane of $\vec{a}$ and $\vec{d}$
  • D
    either $\vec{a}$ or $\vec{b}$ must lie in the plane of $\vec{c}$ and $\vec{d}$

Solution

$((\vec{a} \times \vec{b}) \times(\vec{c} \times \vec{d})) \cdot(\vec{a} \times \vec{d})=0$ $([\vec{a} \vec{c} \vec{d}] \vec{b}-[\vec{b} \vec{c} \vec{d}] \vec{a}) \cdot(\vec{a} \times \vec{d})=0$ $[\vec{a} \vec{c} \vec{d}][\vec{b} \vec{a} \vec{d}]=0$ Either $\vec{c}$ or $\vec{b}$ must lie in the plane of $\vec{a}$ and $\vec{d}$.

Hence, the correct answer is option: 'C'.

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