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

Options

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

Answer

To solve the problem step by step, we will use the properties of vectors and the information given about their resultant magnitudes.

### Step 1: Understand the problem
We are given two vectors whose maximum resultant is 17 units and minimum resultant is 7 units. We need to find the magnitude of the resultant when these two vectors are at right angles to each other.

### Step 2: Define the vectors
Let the magnitudes of the two vectors be \( A \) and \( B \). The maximum and minimum resultant magnitudes can be expressed as:
- Maximum resultant: \( R_{\text{max}} = A + B \)
- Minimum resultant: \( R_{\text{min}} = |A - B| \)

### Step 3: Set up equations
From the problem, we know:
- \( A + B = 17 \) (1)
- \( |A - B| = 7 \) (2)

### Step 4: Solve the equations
From equation (2), we can split it into two cases:
1. \( A - B = 7 \) (Case 1)
2. \( B - A = 7 \) (Case 2)

#### Case 1: \( A - B = 7 \)
From this, we can express \( A \) in terms of \( B \):
\[ A = B + 7 \]

Substituting this into equation (1):
\[ (B + 7) + B = 17 \]
\[ 2B + 7 = 17 \]
\[ 2B = 10 \]
\[ B = 5 \]

Now substituting back to find \( A \):
\[ A = 5 + 7 = 12 \]

#### Case 2: \( B - A = 7 \)
From this, we can express \( B \) in terms of \( A \):
\[ B = A + 7 \]

Substituting this into equation (1):
\[ A + (A + 7) = 17 \]
\[ 2A + 7 = 17 \]
\[ 2A = 10 \]
\[ A = 5 \]

Now substituting back to find \( B \):
\[ B = 5 + 7 = 12 \]

### Step 5: Calculate the resultant for right angles
Now that we have \( A = 12 \) and \( B = 5 \), we can find the resultant when these vectors are at right angles. The formula for the resultant \( R \) of two perpendicular vectors is:
\[ R = \sqrt{A^2 + B^2} \]

Substituting the values:
\[ R = \sqrt{12^2 + 5^2} \]
\[ R = \sqrt{144 + 25} \]
\[ R = \sqrt{169} \]
\[ R = 13 \text{ units} \]

### Conclusion
The magnitude of the resultant of the two vectors when they are at right angles to each other is **13 units**.

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