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Previous Year Question (PYQs)

The value of $\lambda$ for which the volume of the parallelepiped formed by the vectors $\vec i+\lambda\vec j+\vec k,\ \vec j+\lambda\vec k,\ \lambda\vec i+\vec k$ is minimum is




Solution

Volume $=|\det|$ $V=|\lambda^3-3\lambda|$ Minimum when derivative $=0$ $\Rightarrow 3\lambda^2-3=0$ $\Rightarrow \lambda=\pm\dfrac1{\sqrt3}$ Minimum positive value at $\lambda=\dfrac1{\sqrt3}$ Answer: $\boxed{\dfrac1{\sqrt3}}$

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