Question

If $a,b,c\&d$ are the unit vectors such that $(a\times b).(c\times d)=1$and $a.c=\frac{1}{2}$, then:

• A. $a,b,carenon-coplanar$
• B. $a,b,darenon-coplanar$
• C. $b,darenon-parallel$
• D. $a,dareparallelandb,careparallel$

Answer 1

The correct option is C

$b,darenon-parallel$

Finding the value:

Given, $(a\times b).(c\times d)=1$& $a.c=\frac{1}{2}$

Now,

$\left(axb\right).\left(cxd\right)=1,$it is possible when,

$\left|axb\right|.\left|cxd\right|=1\&\left(axc\right)\left|\right|\left(cxd\right)$

Also, $a.c=\frac{1}{2}$& if we say, $b\parallel d$ then,

$\left|cxd\right|\ne 1$

Which is not possible.

Therefore, we can say that b and d are non-parallel.

Hence, correct option is $\left(C\right)$.