Question

Maximum and minimum magnitudes of the resultant of two vectors of magnitudes $P$ and $Q$ are in the ratio $3:1$. Which of the following relations is true?

• A. $P=Q$
• B. $P=2Q$
• C. $PQ=1$
• D. none of the above

Answer 1

The correct option is B $P=2Q$

$\text{Step 1: Resultant of two vectors}$

Let $\theta$ be the angle between two vectors.

Resultant of $P$ and $Q$: $R=\sqrt{P^2+Q^2+2PQ\cos \theta }$

$\text{Step 2: Condition for maximum and minimum resultant}$

For maximum resultant, the vectors are parallel i.e. $\theta =0^{\circ }$

$R_{max}=\sqrt{P^2+Q^2+2PQ\cos 0^o}=\sqrt{(P+Q{)}^2}$

$\Rightarrow R_{max}=P+Q$

For minimum resultant, the vectors are antiparallel i.e. $\theta ={180}^o$

$R_{min}=\sqrt{P^2+Q^2+2PQ\cos {180}^{\circ }}=\sqrt{(P-Q{)}^2}$

$\Rightarrow R_{min}=P-Q$

$\text{Step 3: Calculation}$

Given: $\frac{R_{max}}{R_{min}}=\frac{3}{1}$

$\Rightarrow \frac{P+Q}{P-Q}=\frac{3}{1}$

$\Rightarrow$ $P+Q=3P-3Q$

$\Rightarrow$ $P=2Q$

Hence, option $B$ is correct.