Question

If the volume of a parallelopiped formed by the vectors $\hat{i}+\lambda \hat{j}+\hat{k},\hat{j}+\lambda \hat{k}$ and $\lambda \hat{i}+\hat{k}$ is minimum, then $\lambda$ is equal to :

• A. $-\frac{1}{\sqrt{3}}$
• B. $-\sqrt{3}$
• C. $\frac{1}{\sqrt{3}}$
• D. $\sqrt{3}$

Answer 1

The correct option is C $\frac{1}{\sqrt{3}}$

Volume of parallelopiped
$=\left|\begin{array}{ccc}1& \lambda & 1\\ 0& 1& \lambda \\ \lambda & 0& 1\end{array}\right|=1[1-0]-\lambda [0-{\lambda }^2]+1[0-\lambda ]$
$V=|1+{\lambda }^3-\lambda |$
$\Rightarrow \frac{dV}{d\lambda }=3{\lambda }^2-1=0$
$\Rightarrow \lambda =\pm \frac{1}{\sqrt{3}}$
$\Rightarrow \frac{d^{2}V}{d{\lambda }^2}=6\lambda$
For minimum $V$ $\Rightarrow \frac{d^{2}V}{d{\lambda }^2}>0\text{which is at}\lambda =\frac{1}{\sqrt{3}}$