Question

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

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

Answer 1

The correct option is A $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are non-coplanar

We have, $\overline{a}\overline{b}\overline{c}$ are unit vector,

$(\overline{a}\times \overline{b}).(\overline{c}\times \overline{d})=1$ and $\overline{a}.\overline{c}=\frac{1}{2}$

where as,

$\begin{array}{c}\left|\overline{a}\times \overline{b}\right|=\left|\overline{a}\right|\left|\overline{b}\right|=1\\ \left|\overline{c}\times \overline{d}\right|=\left|\overline{c}\right|\left|\overline{d}\right|=1\end{array}$

Now,

$\begin{array}{c}\left|\overline{a}\times \overline{b}\right|.\left|\overline{c}\times \overline{d}\right|=1\\ \left|\overline{a}\times \overline{b}\right|=\left|\overline{c}\times \overline{d}\right|=1\end{array}$

So,

$\left|\overline{a}\times \overline{b}\right|and\left|\overline{c}\times \overline{d}\right|areparallel$

$\therefore \left|\overline{a}\times \overline{b}\right|=1\to \overrightarrow{a}\perp \overline{b}$ and $\left|\overline{c}\times \overline{d}\right|=1\to \overrightarrow{c}\perp \overline{d}$

The four vector $\overline{a}\times \overline{b}$ is parallel to $\overline{c}\times \overline{d}$

So, $\overline{a}\overline{b}\overline{c}$ are non coplanar.

therefor, the correct option is A.