Question

When two vectors of magnitudes $P$ and $Q$ are inclined at an angle $\Theta$ the magnitude of their resultant $2P$. When the inclination is changed to $180-\Theta$ the magnitude of the resultant is halved. Find the ratio of $P$ to $Q$.

• A. $\sqrt{3}:\sqrt{2}$
• B. $1:\sqrt{3}$
• C. $1:\sqrt{2}$
• D. $\sqrt{2}:\sqrt{3}$

Answer 1

The correct option is D $\sqrt{2}:\sqrt{3}$

According to given condition,
Two vectors of magnitudes $P$ and $Q$ are inclined at an angle $\Theta$ the magnitude of their resultant $2P$.
$R_1^2=(2P{)}^2=P^2+Q^2+2PQ\cos \Theta$
The inclination is changed to $180-\Theta$ the magnitude of the resultant is halved.
$(P{)}^2=P^2+Q^2-2PQ\cos \Theta$
Adding them,
$(5P{)}^2=2P^2+2Q^2\frac{P}{Q}=\frac{\sqrt{2}}{\sqrt{3}}$