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: Sum and difference:
$(5\vec a + 2\vec b) + (\vec a - 3\vec b) = 6\vec a - \vec b$ and $(5\vec a+2\vec b) - (\vec a - 3\vec b) = 4\vec a + 5\vec b$.

Step 2: Compute squared lengths:
Use $|x\vec a + y\vec b|^2 = 8x^2 + 9y^2 + 12xy$. For $(6,-1)$: $288 + 9 - 72 = 225$. For $(4,5)$: $128 + 225 + 240 = 593$.

Step 3: Lengths:
$\sqrt{225} = 15$ and $\sqrt{593}$, option (A).

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