Question

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ and $\overrightarrow{d}$ are unit vectors 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. $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ and $\overrightarrow{d}$ are coplanar vectors
• D. $\overrightarrow{a},\overrightarrow{d}$ are parallel and $\overrightarrow{b},\overrightarrow{c}$ are parallel

Answer 1

The correct option is C $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ and $\overrightarrow{d}$ are coplanar vectors

$(\overline{a}\times \overline{b}).(\overline{c}\times \overline{d})$
$=|\overline{a}\times \overline{b}||\overline{c}\times \overline{d}|cos\theta$
$=1$
Hence
For this to be true,
$(\overline{a}\times \overline{b})||(\overline{c}\times \overline{d})$ and $|\overline{a}\times \overline{b}|=|\overline{c}\times \overline{d}|=1$
Hence
$\overline{a},\overline{b},\overline{c},\overline{d}$ are coplanar vectors.