Question

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ and $\overrightarrow{d}$ are unit vectors such that $(\overrightarrow{a}\times \overrightarrow{b}).(\overrightarrow{c}\times \overrightarrow{d})=\lambda$ and $\overrightarrow{a}.\overrightarrow{c}=\frac{\sqrt{3}}{2}$, then

• A. $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are coplanar if $\lambda =1$
• B. Angle between $\overrightarrow{b}$ and $\overrightarrow{d}$ is ${30}^o$ if $\lambda =-1$
• C. angle between $\overrightarrow{b}$ and $\overrightarrow{d}$ is ${150}^o$ if $\lambda =-1$
• D. If $\lambda =1$ then angle between $\overrightarrow{b}$ and $\overrightarrow{c}$ is ${60}^o$.

Answer 1

The correct option is A $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are coplanar if $\lambda =1$

Now, $a\cdot c=\frac{\sqrt{3}}{2}$ implies a is at an angle of ${30}^{\circ }$ with c .

Since b and d are perpendicular to a and c respectively,

this means the angle between b and d is either ${30}^{\circ }$ or ${90}^{\circ }$ degrees

and therefore b and d are guaranteed not to be parallel.