For what value of a is the volume of the para | vedclass

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(C) Let the vectors be $\vec{\alpha} = \hat{i} + a\hat{j} + \hat{k}$,$\vec{\beta} = \hat{j} + a\hat{k}$,and $\vec{\gamma} = a\hat{i} + \hat{k}$.The vo

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$\frac{1}{\sqrt{3}}$

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(C) Let the vectors be $\vec{\alpha} = \hat{i} + a\hat{j} + \hat{k}$,$\vec{\beta} = \hat{j} + a\hat{k}$,and $\vec{\gamma} = a\hat{i} + \hat{k}$.The volume $V$ of the parallelepiped is given by the absolute value of the scalar triple product: $V = |[\vec{\alpha}, \vec{\beta}, \vec{\gamma}]|$.Calculating the scalar triple product:$[\vec{\alpha}, \vec{\beta}, \vec{\gamma}] = \begin{vmatrix} 1 & a & 1 \ 0 & 1 & a \ a & 0 & 1 \end{vmatrix} = 1(1 - 0) - a(0 - a^2) + 1(0 - a) = 1 + a^3 - a$.Let $f(a) = a^3 - a + 1$. To find the minimum volume,we analyze $f(a)$.$f'(a) = 3a^2 - 1$.Setting $f'(a) = 0$,we get $3a^2 = 1$,so $a = \pm \frac{1}{\sqrt{3}}$.Using the second derivative test: $f''(a) = 6a$.For $a = \frac{1}{\sqrt{3}}$,$f''(\frac{1}{\sqrt{3}}) = 6(\frac{1}{\sqrt{3}}) = 2\sqrt{3} > 0$,which indicates a local minimum.Thus,the volume is minimum at $a = \frac{1}{\sqrt{3}}$.

For what value of $a$ is the volume of the parallelepiped formed by the vectors $\hat{i} + a\hat{j} + \hat{k}$,$\hat{j} + a\hat{k}$,and $a\hat{i} + \hat{k}$ minimum?

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