Question

If $a,b,c,d$ are unit vectors such that $(a\times b).(c\times d)=1$ and $a\cdot c=1/2$ then

• A. $a,b,c$ are non-coplanar
• B. $b,c,d$ are non-coplanar
• C. $b,d$ are non-parallel
• D. $a,d$ are parallel and $b,c$ are parallel

Answer 1

The correct option is C $b,d$ are non-parallel

Let ${\theta }_1$, be the angle between $a$ and $b,{\theta }_2$ be the angle between $c$ and $d$
and $\varphi$ be angle between $a\times b$ and $c\times d$ then
$(a\times b).(c\times d)=1\Rightarrow |a\times b||c\times d|\cos \varphi =1$
$\Rightarrow \sin {\theta }_1\sin {\theta }_2\cos \varphi =1$
$\Rightarrow {\theta }_1=\pi /2,\theta /2=\pi /2,\varphi =0$
As $\varphi =0$ so $a\times b$ and $c\times d.$ are parallel
$\Rightarrow a,b,c,d$ are coplanar Also $a\cdot c=1/2\Rightarrow \cos {\theta }_3=1/2$
(where ${\theta }_3$ is the angle between $a$ and $c$ )
${\theta }_3=\pi /2\Rightarrow$ angle between $a$ and $c$ is $\pi /3$
$\Rightarrow$ angle between $b$ and $d$ is $\pi /3$
$\therefore b$ and $d$ are non-parallel