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# The volume of a parallelopiped formed by $\hat i+a\hat j+\hat k,\:\hat j+a\hat k\:and\:\:a\hat i+\hat k$ is minimum if $a=?$

$(a)\:\:-3\:\:\:\qquad\:\:(b)\:\:3\:\:\:\qquad\:\:(c)\:\:1/\sqrt 3 \:\:\:\qquad\:\:(d)\:\:\sqrt 3$

Toolbox:
• Volume of a parallelopiped formed by $\overrightarrow a,\:\overrightarrow b\:and\:\overrightarrow c$ is given by $[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]$
Given: $\overrightarrow a=\hat i+a\hat j+\hat k,\:\:\overrightarrow b=\hat j+a\hat k,\:\:and\:\:\overrightarrow c=a\hat i+\hat k$
Volume of a parallelopiped formed by $\overrightarrow a,\:\overrightarrow b\:and\:\overrightarrow c$ is given by $[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]$
$V=\left |\begin {array} {ccc} 1 & a & 1 \\0 & 1 & a\\a & 0 & 1\end {array}\right |=1+a^3-a$
Given that V is minimum.
$\Rightarrow\:\large\frac{dV}{da}$$=0\:\:and\:\:\large\frac{d^2V}{da^2}$$=-ve$
$\Rightarrow\:3a^2-1=0\:\:or\:\:a=\pm \large\frac{1}{\sqrt 3}$
$\large\frac{d^2V}{da^2}$$=6a$ is $-ve$ if $a=-\large\frac{1}{\sqrt 3}$