The maximum and minimum magnitudes of the resultant of two vectors are 23 units and 7 units respectively. If these two vectors are acting at right angles to each other, the magnitude of their resultant will be

Options

A. 14
B. 15
C. 16
D. 17

Answer

To find the magnitude of the resultant of two vectors acting at right angles to each other, given their maximum and minimum magnitudes, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Maximum and Minimum Resultants**:
- The maximum resultant \( R_{\text{max}} \) occurs when the two vectors are in the same direction:
\[ R_{\text{max}} = F_1 + F_2 \]
- The minimum resultant \( R_{\text{min}} \) occurs when the two vectors are in opposite directions:
\[ R_{\text{min}} = |F_1 - F_2| \]

2. **Setting Up the Equations**:
- From the problem, we know:
\[ R_{\text{max}} = 23 \quad \text{(units)} \]
\[ R_{\text{min}} = 7 \quad \text{(units)} \]
- Therefore, we can write the equations:
\[ F_1 + F_2 = 23 \quad \text{(1)} \]
\[ |F_1 - F_2| = 7 \quad \text{(2)} \]

3. **Solving the Equations**:
- From equation (2), we can consider two cases:
- Case 1: \( F_1 - F_2 = 7 \)
- Case 2: \( F_2 - F_1 = 7 \)

- **Case 1**: \( F_1 - F_2 = 7 \)
- Adding equations (1) and (2):
\[ (F_1 + F_2) + (F_1 - F_2) = 23 + 7 \]
\[ 2F_1 = 30 \implies F_1 = 15 \quad \text{(units)} \]
- Substituting \( F_1 \) back into equation (1):
\[ 15 + F_2 = 23 \implies F_2 = 8 \quad \text{(units)} \]

- **Case 2**: \( F_2 - F_1 = 7 \)
- Adding equations (1) and (2):
\[ (F_1 + F_2) + (F_2 - F_1) = 23 + 7 \]
\[ 2F_2 = 30 \implies F_2 = 15 \quad \text{(units)} \]
- Substituting \( F_2 \) back into equation (1):
\[ F_1 + 15 = 23 \implies F_1 = 8 \quad \text{(units)} \]

- In both cases, we find:
\[ F_1 = 15 \quad \text{(units)}, \quad F_2 = 8 \quad \text{(units)} \]

4. **Finding the Resultant Magnitude**:
- Since the vectors are acting at right angles, the magnitude of the resultant \( R \) is given by:
\[ R = \sqrt{F_1^2 + F_2^2} \]
- Substituting the values:
\[ R = \sqrt{15^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} \]
- Therefore:
\[ R = 17 \quad \text{(units)} \]

### Final Answer:
The magnitude of the resultant of the two vectors is **17 units**.