A parallelogram is constructed on the vectors \(2\vec{a}+\vec{b}\) and \(\vec{a}-2\vec{b}\) as adjacent sides, where \(|\vec{a}| = 2\), \(|\vec{b}| = \sqrt{2}\) and the angle between \(\vec{a}\) and \(\vec{b}\) is \(\displaystyle\frac{\pi}{4}\). The length of the longer diagonal of the parallelogram is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\sqrt{34}\)
The diagonals of a parallelogram are the sum and the difference of its adjacent sides:
\(\vec{d}_1 = (2\vec{a}+\vec{b})+(\vec{a}-2\vec{b}) = 3\vec{a}-\vec{b}\)
\(\vec{d}_2 = (2\vec{a}+\vec{b})-(\vec{a}-2\vec{b}) = \vec{a}+3\vec{b}\)
The scalar product of \(\vec{a}\) and \(\vec{b}\) is
\( \displaystyle \vec{a}\cdot\vec{b} = 2\times\sqrt{2}\times\cos\frac{\pi}{4} = 2 \)
The squares of the lengths are
\(|\vec{d}_1|^2 = 9|\vec{a}|^2-6\,\vec{a}\cdot\vec{b}+|\vec{b}|^2 = 36-12+2 = 26\)
\(|\vec{d}_2|^2 = |\vec{a}|^2+6\,\vec{a}\cdot\vec{b}+9|\vec{b}|^2 = 4+12+18 = 34\)
So the longer diagonal has length \(\sqrt{34}\).
Why the other options are incorrect
Option (1) \(\sqrt{26}\) — Incorrect. It is the length of the shorter diagonal \(3\vec{a}-\vec{b}\).
Option (2) \(\sqrt{38}\) — Incorrect. \(36+2 = 38\) leaves out the term \(-6\,\vec{a}\cdot\vec{b}\), as if \(\vec{a}\) and \(\vec{b}\) were perpendicular.
Option (4) \(\sqrt{50}\) — Incorrect. \(36+12+2 = 50\) is \(|3\vec{a}+\vec{b}|^2\); the diagonal is \(3\vec{a}-\vec{b}\), whose middle term is negative.