Question

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be non-coplanar unit vectors equally inclined to one another at an acute angle $\theta$. Then $[\vec{a}\, \vec{b}\, \vec{c}]$ 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. $(1-\cos \theta) \sqrt{1+2 \cos \theta}$
• D. None of these

Answer 1

Answer: (C)

$|\vec{a}|=|\vec{b}|=|\vec{c}|=1$
$\vec{a} \cdot \vec{b}=(1)(1) \cos \theta=\cos \theta$ and
$\vec{c} \cdot \vec{a}=\cos \theta, \vec{b} \cdot \vec{c}=\cos \theta$
$\vec{a} \, \vec{b} \, \vec{c}^{2}=\begin{vmatrix}\vec{a} \cdot \vec{a} & \vec{a} \cdot \vec{b} & \vec{a} \cdot \vec{c} \\ \vec{b} \cdot \vec{a} & \vec{b} \cdot \vec{b} & \vec{b} \cdot \vec{c} \\ \vec{c} \cdot \vec{a} & \vec{c} \cdot \vec{b} & \vec{c} \cdot \vec{c}\end{vmatrix}=\begin{vmatrix}1 & \cos \theta & \cos \theta \\ \cos \theta & 1 & \cos \theta \\ \cos \theta & \cos \theta & 1\end{vmatrix}$
Operate $R_{1} \rightarrow R_{1}+R_{2}+R_{3}$
$=(1+2 \cos \theta)\begin{vmatrix}1 & 1 & 1 \\ \cos \theta & 1 & \cos \theta \\ \cos \theta & \cos \theta & 1\end{vmatrix}$
Operate $C_{2} \rightarrow C_{2}-C_{1} ; C_{3} \rightarrow C_{3}-C_{1}$
$=(1+2 \cos \theta)\begin{vmatrix}1 & 0 & 0 \\ \cos \theta & 1-\cos \theta & 0 \\ \cos \theta & 0 & 1-\cos \theta\end{vmatrix}$
$=(1+2 \cos \theta)(1-\cos \theta)^{2}$
$\therefore [\vec{a} \,\,\vec{b}\,\, \vec{c}]=(1-\cos \theta) \sqrt{1+2 \cos \theta}$