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The value of ' \(a\) ' for which the scalar triple product formed by the vectors \(\vec{a}=\hat{i}+a \hat{j}+\hat{k}, \vec{\beta}=\hat{j}+a \hat{k}\) and \(\vec{\gamma}=a \hat{i}+\hat{k}\) is maximum, is
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The correct answer is:
\(-\frac{1}{\sqrt{3}}\)
Hint : \(\Delta=1+\mathrm{a}^3-\mathrm{a}\) For \(\max ; \frac{\mathrm{d} \Delta}{\mathrm{da}}=0\)
\(\frac{\mathrm{d} \Delta}{\mathrm{da}}=3 \mathrm{a}^2-1 \quad \mathrm{a}= \pm \frac{1}{\sqrt{3}}\)
\(\frac{\mathrm{d}^2 \Delta}{\mathrm{da}^2}=6 \mathrm{a} < 0\) for \(\mathrm{a}=-\frac{1}{\sqrt{3}}\)
\(\frac{\mathrm{d} \Delta}{\mathrm{da}}=3 \mathrm{a}^2-1 \quad \mathrm{a}= \pm \frac{1}{\sqrt{3}}\)
\(\frac{\mathrm{d}^2 \Delta}{\mathrm{da}^2}=6 \mathrm{a} < 0\) for \(\mathrm{a}=-\frac{1}{\sqrt{3}}\)
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