Mathematics | VITEEE 2017 | Discussion Page

If $\left(\left(\vec{a}\times\vec{b}\right)\times\left(\vec{c}\times\vec{d}\right)\right).\left(\vec{a}\times\vec{d}\right)=0$

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 $\vec{c}$

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 $\vec{d}$

Option C

$\left(\left(\vec{a}\times\vec{b}\right)\times\left(\vec{c}\times\vec{d}\right)\right).\left(\vec{a}\times\vec{d}\right)=0$

$\left(\left[\vec{a}\vec{c}\vec{d}\right]\vec{b} -\left[\vec{b}\vec{c}\vec{d}\right]\vec{a}\right).\left(\vec{a}\times\vec{d}\right)=0$

$\left[\vec{a}\vec{c}\vec{d}\right]\left[\vec{b}\vec{a}\vec{d}\right]=0$

Either $\vec{c}$ or $\vec{b}$ must lie in the plane of $\vec{a}$ and $\vec{d}$

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