A parallelogram is constructed on \(5\vec{a} + 2\vec{b}\) and \(\vec{a} - 3\vec{b}\) where |a| = 2\(\sqrt{2}\) and |b| = 3. If the angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{\pi}{4}\), then what is the length of the longer diagonal ?

\(\sqrt{593}\)

The vector representing one of the diagonals is \(5\vec{a} + 2\vec{b} + \vec{a} - 3\vec{b} = 6\vec{a} - \vec{b}\)

Thus, the length of the diagonal = \(\sqrt{(6\vec{a}-\vec{b})(6\vec{a}-\vec{b})}\)

= \(\sqrt{36|\vec{a}|^2 + |\vec{b}|^2 - 12\vec{a}\vec{b}} = \sqrt{36 * 8 + 9 - 12 * 3 * 2} = 15 \)

The other diagonal is \(5\vec{a} + 2\vec{b} - \vec{a} - 3\vec{b} = 4\vec{a} + 5\vec{b}\)

Its length = \(\sqrt{16|\vec{a}|^2 + 25|\vec{b}|^2 + 40\vec{a}\vec{b}}\)

= \(\sqrt{128 + 225 + 40 * 2 * 3} \)

= \(\sqrt{593} \)