The volume of parallelopiped formed by vectors $\hat{i}+m \hat{j}+\hat{k}, \hat{j}+m \hat{k}$ and $m \hat{i}+\hat{k}$ becomes minimum when $m$ is

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Aicharya · Accepted Answer

Volume of the parallelopiped formed by vectors is i.e., $V=\left|\begin{array}{ccc}1 & m & 1 \\ 0 & 1 & m \\ m & 0 & 1\end{array}\right|=1-m+m^3$ $\therefore \quad \frac{d V}{d m}=-1+3 m^2, \frac{d^2 V}{d a^2}=6 m$
For max. or $\min$. of $V, \frac{d V}{d m}=0$ $\therefore \quad \mathrm{m}^2=\frac{1}{3}$ $\therefore \quad \mathrm{m}=\frac{1}{\sqrt{3}}$ $\frac{d^2 V}{d m^2}=6 m\gt0$ for $m=\frac{1}{\sqrt{3}}$ $\therefore \quad \mathrm{V}$ is minimum for $\mathrm{m}=\frac{1}{\sqrt{3}}$

The volume of parallelopiped formed by vectors $\hat{i}+m \hat{j}+\hat{k}, \hat{j}+m \hat{k}$ and $m…

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