Question

If a parallelogram is constructed on vectors \(\vec{a}=3\vec{p}-\vec{q}\), \(\vec{b}=\vec{p}+3\vec{q}\) and \(|\vec{p}|=|\vec{q}|=2\) and the angle between \(\vec{p}\) and \(\vec{q}\) is \(\frac{\pi }{3}\), then \(|\vec{a}|:|\vec{b}|=\ldots\)

• A. \(\sqrt{5}:\sqrt{13}\)
• B. \(\sqrt{7}:\sqrt{13}\)
• C. \(\sqrt{13}:\sqrt{7}\)
• D. \(\sqrt{13}:\sqrt{5}\)

Hint:

Find p dot q first, then compute the squared magnitudes of a and b.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
\(\vec{p}\cdot\vec{q}=|\vec{p}||\vec{q}|\cos\dfrac{\pi}{3}=2\times2\times\dfrac{1}{2}=2\), and \(|\vec{p}|^{2}=|\vec{q}|^{2}=4\).

Step 2: Magnitude of a
\(|\vec{a}|^{2}=9|\vec{p}|^{2}+|\vec{q}|^{2}-6\,\vec{p}\cdot\vec{q}=36+4-12=28\).

Step 3: Magnitude of b
\(|\vec{b}|^{2}=|\vec{p}|^{2}+9|\vec{q}|^{2}+6\,\vec{p}\cdot\vec{q}=4+36+12=52\).

Step 4: Ratio
\(|\vec{a}|:|\vec{b}|=\sqrt{28}:\sqrt{52}=2\sqrt{7}:2\sqrt{13}=\sqrt{7}:\sqrt{13}\), option (B).

Final Answer:
The ratio is root 7 to root 13. \[ \boxed{\text{(B) }\sqrt{7}:\sqrt{13}} \]