The maximum and minimum magnitude of the resultant of two vectors are 17 units and 7 units respectively. Then the magnitude of the resultant of the vectors when they act perpendicular to each other is :

Options

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

Explanation

To find the magnitude of the resultant of two vectors when they act perpendicular to each other, given their maximum and minimum resultant magnitudes, we can follow these steps:

### Step 1: Understand the given information
We know:
- The maximum magnitude of the resultant \( R_{max} = 17 \) units.
- The minimum magnitude of the resultant \( R_{min} = 7 \) units.

### Step 2: Set up equations for the vectors
Let the magnitudes of the two vectors be \( A \) and \( B \).
- The maximum resultant occurs when the vectors are in the same direction:
\[ R_{max} = A + B = 17 \quad \text{(Equation 1)} \]
- The minimum resultant occurs when the vectors are in opposite directions:
\[ R_{min} = A - B = 7 \quad \text{(Equation 2)} \]

### Step 3: Solve the equations
Add Equation 1 and Equation 2:
\[ (A + B) + (A - B) = 17 + 7 \]
This simplifies to:
\[ 2A = 24 \implies A = 12 \]

Now, substitute \( A = 12 \) into Equation 1 to find \( B \):
\[ 12 + B = 17 \implies B = 17 - 12 = 5 \]

### Step 4: Calculate the resultant when vectors are perpendicular
When two vectors \( A \) and \( B \) act at right angles (90 degrees) to each other, the magnitude of the resultant \( R \) can be calculated using the Pythagorean theorem:
\[ R = \sqrt{A^2 + B^2} \]
Substituting the values of \( A \) and \( B \):
\[ R = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]

### Conclusion
The magnitude of the resultant of the vectors when they act perpendicular to each other is \( 13 \) units.

---