Question

If a parallelogram is constructed on the vectors \(\overset{̄}{a} = 3\overset{̄}{p}-\overset{̄}{q},\overset{̄}{b} = \overset{̄}{p}+3\overset{̄}{q}\) and \(|\overset{̄}{p}| = 3,|\overset{̄}{q}| = 2\) and angle between \(\overset{̄}{p}\) and \(\overset{̄}{q}\) is \(\frac{π}{3}\), then the ratio of the lengths of adjacent sides \(\overset{̄}{a}\) and \(\overset{̄}{b}\) of the parallelogram is

• A. \(\sqrt{57}:\sqrt{54}\)
• B. \(\sqrt{67}:\sqrt{63}\)
• C. \(\sqrt{63}:\sqrt{47}\)
• D. \(\sqrt{57}:\sqrt{52}\)

Hint:

Find the square of each side using the dot product with p dot q = 3.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The length of a vector squared is the dot product with itself. For this we need \(\bar{p}\cdot\bar{q} = |\bar{p}||\bar{q}|\cos\dfrac{\pi}{3} = 3\times 2\times\dfrac{1}{2} = 3\).

Step 2: Length of \(\bar{a}\).
\[ |\bar{a}|^2 = |3\bar{p} - \bar{q}|^2 = 9|\bar{p}|^2 - 6\,\bar{p}\cdot\bar{q} + |\bar{q}|^2 = 81 - 18 + 4 = 67 \]

Step 3: Length of \(\bar{b}\).
\[ |\bar{b}|^2 = |\bar{p} + 3\bar{q}|^2 = |\bar{p}|^2 + 6\,\bar{p}\cdot\bar{q} + 9|\bar{q}|^2 = 9 + 18 + 36 = 63 \]

Step 4: Ratio.
\(|\bar{a}| : |\bar{b}| = \sqrt{67} : \sqrt{63}\), which is option (B).

Final Answer:
The ratio of the sides is \(\sqrt{67} : \sqrt{63}\). \[ \boxed{\sqrt{67} : \sqrt{63}} \]