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 solve the problem step by step, we will use the information provided about the maximum and minimum magnitudes of the resultant of two vectors.

### Step 1: Understand the Problem
We are given two vectors, A and B, with the maximum resultant \( R_{max} = 17 \) units and the minimum resultant \( R_{min} = 7 \) units. We need to find the magnitude of the resultant when these two vectors are at right angles to each other.

### Step 2: Set Up the Equations
The maximum resultant occurs when the two vectors are in the same direction:
\[ R_{max} = A + B \]
From the problem, we know:
\[ A + B = 17 \quad \text{(Equation 1)} \]

The minimum resultant occurs when the two vectors are in opposite directions:
\[ R_{min} = |A - B| \]
From the problem, we know:
\[ A - B = 7 \quad \text{(Equation 2)} \]

### Step 3: Solve the Equations
Now we will solve these two equations simultaneously.

1. From Equation 1:
\[ A + B = 17 \]
2. From Equation 2:
\[ A - B = 7 \]

We can add these two equations:
\[ (A + B) + (A - B) = 17 + 7 \]
This simplifies to:
\[ 2A = 24 \implies A = 12 \]

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

### Step 4: Calculate the Resultant When Vectors are Perpendicular
When the vectors A and B are at right angles (90 degrees), the magnitude of the resultant \( R \) is given by:
\[ R = \sqrt{A^2 + B^2} \]
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 two vectors when they are at right angles to each other is \( 13 \) units.