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 relation is true?

Options

A. `P=2Q`
B. `P=Q`
C. `PQ=1`
D. none of these

Explanation

To solve the problem, we need to analyze the relationship between the maximum and minimum magnitudes of the resultant of two vectors \( P \) and \( Q \) given that their ratio is \( 3:1 \).

### Step-by-Step Solution:

1. **Understanding Resultant Vectors**:
The resultant \( R \) of two vectors \( P \) and \( Q \) can be expressed as:
\[ R = \sqrt{P^2 + Q^2 + 2PQ \cos \theta} \]
where \( \theta \) is the angle between the two vectors.

2. **Maximum and Minimum Values of Resultant**:
- The **maximum** value of \( R \) occurs when \( \theta = 0^\circ \) (vectors are in the same direction):
\[ R_{\text{max}} = P + Q \]
- The **minimum** value of \( R \) occurs when \( \theta = 180^\circ \) (vectors are in opposite directions):
\[ R_{\text{min}} = |P - Q| = P - Q \quad \text{(assuming \( P > Q \))} \]

3. **Setting Up the Ratio**:
According to the problem, the ratio of the maximum to minimum resultant is given as:
\[ \frac{R_{\text{max}}}{R_{\text{min}}} = \frac{3}{1} \]
Substituting the expressions for \( R_{\text{max}} \) and \( R_{\text{min}} \):
\[ \frac{P + Q}{P - Q} = 3 \]

4. **Cross-Multiplying**:
Cross-multiplying gives:
\[ P + Q = 3(P - Q) \]

5. **Expanding the Equation**:
Expanding the right side:
\[ P + Q = 3P - 3Q \]

6. **Rearranging the Equation**:
Rearranging the terms to isolate \( P \) and \( Q \):
\[ P + Q + 3Q = 3P \]
\[ P + 4Q = 3P \]
\[ 4Q = 3P - P \]
\[ 4Q = 2P \]

7. **Final Relation**:
Dividing both sides by 2 gives:
\[ 2Q = P \]
or
\[ P = 2Q \]

### Conclusion:
Thus, the relation that holds true is:
\[ P = 2Q \]