Question

Let $\vec{a}\times (\vec{b}\times \vec{c})=\frac{\vec{b}}{3}+\frac{\vec{c}}{2}$ and $\vec{b}\times (\vec{c}\times \vec{a})=-\frac{\vec{c}}{2}$. If $\vec{a},\vec{b}$ and $\vec{c}$ are non-collinear pairwise unit vectors, then volume of a parallelepiped, whose coterminous edges are $\vec{a},\vec{b}$ and $\vec{c}$, is

• A. $\frac{11}{36}$
• B. $\frac{1}{2}$
• C. $\frac{23}{36}$
• D. None of these

Answer 1

Answer: (D)

$\vec{a}\times (\vec{b}\times \vec{c})=\frac{\vec{b}}{3}+\frac{\vec{c}}{2}$
$\Rightarrow (\vec{a}\cdot \vec{c})\vec{b}-(\vec{a}\cdot \vec{b})\vec{c}=\frac{\vec{b}}{3}+\frac{\vec{c}}{2}$
$\Rightarrow \vec{a}\cdot \vec{c}=\frac{1}{3}$ and $\vec{a}\cdot \vec{b}=-\frac{1}{2}$
Also $\vec{b}\times (\vec{c}\times \vec{a})=-\frac{\vec{c}}{2}$
$\Rightarrow (\vec{b}\cdot \vec{a})\vec{c}-(\vec{b}\cdot \vec{c})\vec{a}=-\frac{\vec{c}}{2}$
$\Rightarrow \vec{b}\cdot \vec{c}=0$
Volume of parallelepiped $=|[\vec{a}\vec{b}\vec{c}]|$
${\left[\begin{array}{ccc}\vec{a}& \vec{b}& \vec{c}\end{array}\right]}^2=\left|\begin{array}{ccc}\vec{a}\cdot \vec{a}& \vec{a}\cdot \vec{b}& \vec{a}\cdot \vec{c}\\ \vec{b}\cdot \vec{a}& \vec{b}\cdot \vec{b}& \vec{b}\cdot \vec{c}\\ \vec{c}\cdot \vec{a}& \vec{c}\cdot \vec{b}& \vec{c}\cdot \vec{c}\end{array}\right|$
$=\left|\begin{array}{ccc}1& -\frac{1}{2}& \frac{1}{3}\\ -\frac{1}{2}& 1& 0\\ \frac{1}{3}& 0& 1\end{array}\right|=\frac{23}{36}$
Volume $=\frac{\sqrt{23}}{6}$