The value of $a$ for which the volume of the parallelopiped formed by the vectors $\hat i+a\hat j+\hat k,\:\:\hat j+a\hat k,\:\:a\hat i+\hat k$ is minimum is ?

Answer 1

Volume of the parallelopiped (V) $=\left |\begin {array} {ccc}1 & a & 1 \\0 & 1 & a\\a & 0 &1\end {array}\right|$

$\Rightarrow\:V=1+a^3-a$

If $Volume$ is minimum, then $\large\frac{dV}{da}$$=0\:and\:\large\frac{d^2V}{da^2}$ is $+ve$

$\Rightarrow\:3a^2-1=0\:\:and\:\: 6a$ is $+ve$.

$\Rightarrow\:a=\large\frac{1}{\sqrt 3}$ if $V$ is minimum.