Question:
Let O be the origin and let the vectors \(\overrightarrow{OA} = -3\hat{i} + 7\hat{j} + 5\hat{k}\), \(\overrightarrow{OB} = -5\hat{i} + 7\hat{j} - 3\hat{k}\) and \(\overrightarrow{OC} = \hat{u}\) represent three sides of a parallelopiped, where \(\hat{u}\) is a unit vector in the \(xy\)- plane. If the maximum volume of the parallelopiped is \(2\sqrt{\alpha}\), then \(\alpha\) is equal to ________.
Let O be the origin and let the vectors \(\overrightarrow{OA} = -3\hat{i} + 7\hat{j} + 5\hat{k}\), \(\overrightarrow{OB} = -5\hat{i} + 7\hat{j} - 3\hat{k}\) and \(\overrightarrow{OC} = \hat{u}\) represent three sides of a parallelopiped, where \(\hat{u}\) is a unit vector in the \(xy\)- plane. If the maximum volume of the parallelopiped is \(2\sqrt{\alpha}\), then \(\alpha\) is equal to ________.
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Volume \(=|\hat u\cdot(\overrightarrow{OA}\times\overrightarrow{OB})|\); maximise over the direction of \(\hat u\) in the xy-plane.
Updated On: Nov 12, 2026
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Correct Answer: 1073
Solution and Explanation
\(\overrightarrow{OA}\times\overrightarrow{OB}=(7\cdot(-3)-5\cdot7)\hat i-((-3)(-3)-5(-5))\hat j+((-3)(7)-7(-5))\hat k=-56\hat i-34\hat j+14\hat k\).
Let \(\hat u=\cos\theta\,\hat i+\sin\theta\,\hat j\). Volume \(V=|-56\cos\theta-34\sin\theta|=|56\cos\theta+34\sin\theta|\).
Maximum \(V=\sqrt{56^2+34^2}=\sqrt{3136+1156}=\sqrt{4292}=\sqrt{4\cdot1073}=2\sqrt{1073}\).
Comparing with \(2\sqrt\alpha\): \(\alpha=1073\).
Answer: 1073
Let \(\hat u=\cos\theta\,\hat i+\sin\theta\,\hat j\). Volume \(V=|-56\cos\theta-34\sin\theta|=|56\cos\theta+34\sin\theta|\).
Maximum \(V=\sqrt{56^2+34^2}=\sqrt{3136+1156}=\sqrt{4292}=\sqrt{4\cdot1073}=2\sqrt{1073}\).
Comparing with \(2\sqrt\alpha\): \(\alpha=1073\).
Answer: 1073
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