The maximum and minimum magnitude of the resultant of two given vectors are 17 units and 7 unit respectively. If these two vectors are at right angles to each other, the magnitude of their resultant is

Options

A. 14
B. 16
C. 18
D. 13

Explanation

To find the magnitude of the resultant of two vectors that are at right angles to each other, we can use the Pythagorean theorem. Given the maximum and minimum magnitudes of the resultant are 17 units and 7 units respectively, we can derive the magnitudes of the individual vectors.

### Step-by-Step Solution:

1. **Understanding the Maximum and Minimum Resultants:**
- The maximum resultant \( R_{max} \) occurs when the two vectors are in the same direction:
\[ R_{max} = A + B = 17 \text{ units} \]
- The minimum resultant \( R_{min} \) occurs when the two vectors are in opposite directions:
\[ R_{min} = |A - B| = 7 \text{ units} \]

2. **Setting Up the Equations:**
- From the maximum resultant:
\[ A + B = 17 \quad (1) \]
- From the minimum resultant:
\[ |A - B| = 7 \quad (2) \]

3. **Solving the Equations:**
- From equation (2), we can write two cases:
- Case 1: \( A - B = 7 \)
- Case 2: \( B - A = 7 \) (which is not possible since \( A + B = 17 \) implies \( A > B \))

- Therefore, we have:
\[ A - B = 7 \quad (3) \]

4. **Adding Equations (1) and (3):**
- Adding equations (1) and (3):
\[ (A + B) + (A - B) = 17 + 7 \]
\[ 2A = 24 \implies A = 12 \]

5. **Substituting to Find B:**
- Now substitute \( A = 12 \) back into equation (1):
\[ 12 + B = 17 \implies B = 5 \]

6. **Calculating the Resultant:**
- Since the vectors \( A \) and \( B \) are at right angles, we use the Pythagorean theorem to find the resultant \( R \):
\[ R = \sqrt{A^2 + B^2} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \text{ units} \]

### Final Answer:
The magnitude of the resultant of the two vectors at right angles to each other is **13 units**.