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 vector \(\vec{A}\)**:
- 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 vector \(\vec{A} + \vec{B}\)**:
- 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. **Express vector \(\vec{B}\)**:
- We can express \(\vec{B}\) in terms of \(\vec{A}\) and \(\vec{A} + \vec{B}\):
\[ \vec{B} = (\vec{A} + \vec{B}) - \vec{A} \]
- Substituting the known values:
\[ \vec{B} = (11 \hat{i} + 9 \hat{j}) - (7 \hat{i} + 6 \hat{j}) \]

4. **Calculate the components of vector \(\vec{B}\)**:
- For the x-component:
\[ B_x = 11 - 7 = 4 \]
- For the y-component:
\[ B_y = 9 - 6 = 3 \]
- Therefore, we can write:
\[ \vec{B} = 4 \hat{i} + 3 \hat{j} \]

5. **Calculate the magnitude of vector \(\vec{B}\)**:
- The magnitude of vector \(\vec{B}\) can be calculated using the formula:
\[ |\vec{B}| = \sqrt{B_x^2 + B_y^2} \]
- Substituting the values:
\[ |\vec{B}| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

6. **Final Answer**:
- The magnitude of vector \(\vec{B}\) is:
\[ |\vec{B}| = 5 \]