Question

A parallelogram is constructed on \(5\overset{̄}{a}+2\overset{̄}{b}\) and \(\overset{̄}{a}-3\overset{̄}{b}\) as its adjacent sides, with \(|\overset{̄}{a}| = 2\sqrt{2},|\overset{̄}{b}| = 3\) . The angle between \(\overset{̄}{a}\) and \(\overset{̄}{b}\) is \(\frac{π}{4}\) . Then the length of the diagonals of the parallelogram are

• A. \(15,\sqrt{593}\)
• B. \(15, 593\)
• C. \(225, 593\)
• D. \(20, 593\)

Hint:

The diagonals are the sum and the difference of the adjacent sides.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
For adjacent sides \(\vec u\) and \(\vec v\), the diagonals are \(\vec u + \vec v\) and \(\vec u - \vec v\).

Step 2: Dot product of a and b:
\[ \vec a\cdot\vec b = 2\sqrt2\cdot3\cos\frac\pi4 = 6\sqrt2\cdot\frac{1}{\sqrt2} = 6 \]
\(|\vec a|^2 = 8\), \(|\vec b|^2 = 9\).

Step 3: First diagonal:
\(\vec u + \vec v = 6\vec a - \vec b\):
\[ |6\vec a - \vec b|^2 = 36(8) + 9 - 12(6) = 288 + 9 - 72 = 225 \Rightarrow 15 \]

Step 4: Second diagonal:
\(\vec u - \vec v = 4\vec a + 5\vec b\):
\[ |4\vec a + 5\vec b|^2 = 16(8) + 25(9) + 40(6) = 128 + 225 + 240 = 593 \Rightarrow \sqrt{593} \]
The lengths are \(15\) and \(\sqrt{593}\), option (A). Options (B), (C), (D) give squared values instead of lengths.

Final Answer:
The diagonals are 15 and root 593. \[ \boxed{\text{(A) }15,\ \sqrt{593}} \]