Question

Let $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ br non-coplanar unit vectors equally inclined to one another at an acute angle $\theta$. Then $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]$ in terms of $\theta$ is equal to

• A. $(1+\cos \theta )\sqrt{\cos 2\theta }$
• B. $(1+\cos \theta )\sqrt{1-2\cos 2\theta }$
• C. $(1-\cos \theta )\sqrt{1+2\cos 2\theta }$
• D. $noneofthese$

Answer 1

Answer: (C)

$(1-\cos \theta )\sqrt{1+2\cos 2\theta }$

for $\theta ={90}^{\circ }$ it becomes a cube of volume 1

$(1-\cos \theta )\sqrt{1+2\cos 2\theta }$ and $(1+\cos \theta )\sqrt{1-2\cos 2\theta }$

satisfy it.

for $\theta =0^{\circ }$ volume is 0

Option C satisfies it

$\therefore \left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]=(1-\cos \theta )\sqrt{1+2\cos 2\theta }$