The value of a for which the volume of parallelepiped formed by $\hat{i}+a \hat{j}+\hat{k}, \hat{j}+a…
The value of a for which the volume of parallelepiped formed by $\hat{i}+a \hat{j}+\hat{k}, \hat{j}+a \hat{k}$ and $a \hat{i}+\hat{k}$ becomes minimum is
$\frac{-1}{\sqrt{3}}$
$\frac{1}{\sqrt{3}}$
$\sqrt{3}$
$-\sqrt{3}$
Solution
i.e., $V=\left|\begin{array}{lll}1 & a & 1 \\ 0 & 1 & a \\ a & 0 & 1\end{array}\right| \doteq 1-a+a^3$
$\therefore \quad \frac{d V}{d a}=-1+3 a^2, \frac{d^2 V}{d a^2}=6 a$
For max. or min. of $V, \frac{d V}{d a}=0$
$\therefore \quad \mathrm{a}^2=\frac{1}{3}$
$\therefore \quad a=\frac{1}{\sqrt{3}}$.
$\frac{d^2 V}{d a^2}=6 a\gt0$ for $a=\frac{1}{\sqrt{3}}$
$\therefore \quad \mathrm{V}$ is minimum for $\mathrm{a}=\frac{1}{\sqrt{3}}$