Question

If the volume of parallelopiped formed by the vectors \(\hat{i}+\lambda\hat{j}+\hat{k}\), \(\hat{j}+\lambda\hat{k}\) and \(\lambda\hat{i}+\hat{k}\) is minimum, then \(\lambda\) is equal to :

• A. \(-\frac{1}{\sqrt{3}}\)
• B. \(\sqrt{3}\)
• C. \(-\sqrt{3}\)
• D. \(\frac{1}{\sqrt{3}}\)

Hint:

Volume is the scalar triple product \(\lambda^3-\lambda+1\); find where its derivative vanishes and the second derivative is positive.

Answer 1

The Correct Option is D

Volume \(V=\begin{vmatrix}1&\lambda&1\\0&1&\lambda\\\lambda&0&1\end{vmatrix}\).
Expanding along the first column: \(1(1-0)-0+\lambda(\lambda^2-1)=\lambda^3-\lambda+1\).
\(V'=3\lambda^2-1=0\Rightarrow\lambda=\pm\frac1{\sqrt3}\); \(V''=6\lambda\).
\(V''\gt0\) only for \(\lambda=\frac{1}{\sqrt3}\), so the minimum is at \(\lambda=\frac1{\sqrt3}\) (there \(V=1-\frac{2}{3\sqrt3}\approx0.615\), which is positive).
Answer: (4) \(\frac{1}{\sqrt{3}}\)