The magnitude of x and y components of `vec(A)` are 7 and 6 respectively . Also , the magnitudes of x and y components of `vec(A)+vec(B)` are 11 and 9 respectively . Calculate the magnitude of vector `vec(B)`

Options

A. 10
B. 5
C. 6
D. 3

Answer

To solve the problem, we need to find the magnitude of vector \(\vec{B}\) given the components of vectors \(\vec{A}\) and \(\vec{A} + \vec{B}\).

### Step-by-Step Solution:

1. **Identify the Components of \(\vec{A}\)**:
- The x-component of \(\vec{A}\) is given as 7.
- The y-component of \(\vec{A}\) is given as 6.
- Therefore, we can write \(\vec{A} = 7 \hat{i} + 6 \hat{j}\).

2. **Identify the Components of \(\vec{A} + \vec{B}\)**:
- The x-component of \(\vec{A} + \vec{B}\) is given as 11.
- The y-component of \(\vec{A} + \vec{B}\) is given as 9.
- Therefore, we can write \(\vec{A} + \vec{B} = 11 \hat{i} + 9 \hat{j}\).

3. **Set Up the Equations**:
- The x-component of \(\vec{A} + \vec{B}\) can be expressed as:
\[ 7 + x = 11 \]
where \(x\) is the x-component of \(\vec{B}\).
- The y-component of \(\vec{A} + \vec{B}\) can be expressed as:
\[ 6 + y = 9 \]
where \(y\) is the y-component of \(\vec{B}\).

4. **Solve for \(x\) and \(y\)**:
- From the first equation:
\[ x = 11 - 7 = 4 \]
- From the second equation:
\[ y = 9 - 6 = 3 \]

5. **Write the Components of \(\vec{B}\)**:
- Now we have:
\[ \vec{B} = 4 \hat{i} + 3 \hat{j} \]

6. **Calculate the Magnitude of \(\vec{B}\)**:
- The magnitude of vector \(\vec{B}\) is calculated using the formula:
\[ |\vec{B}| = \sqrt{x^2 + y^2} \]
- Substituting the values of \(x\) and \(y\):
\[ |\vec{B}| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Final Answer:
The magnitude of vector \(\vec{B}\) is \(5\) units.