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 $=|[\vec{a}\,\, \vec{b}\,\, \vec{c}]|$
$\begin{bmatrix}\vec{a} & \vec{b} & \vec{c}\end{bmatrix}^{2}=\begin{vmatrix}\vec{a} \cdot \vec{a} & \vec{a} \cdot \vec{b} & \vec{a} \cdot \vec{c} \\ \vec{b} \cdot \vec{a} & \vec{b} \cdot \vec{b} & \vec{b} \cdot \vec{c} \\ \vec{c} \cdot \vec{a} & \vec{c} \cdot \vec{b} & \vec{c} \cdot \vec{c}\end{vmatrix}$
$=\begin{vmatrix}1 & -\frac{1}{2} & \frac{1}{3} \\ -\frac{1}{2} & 1 & 0 \\ \frac{1}{3} & 0 & 1\end{vmatrix}=\frac{23}{36}$
Volume $=\frac{\sqrt{23}}{6}$