`vec(a)=5` units due South West
`vec(b)=5` units due `53^(@)` North of East.
`vec(c)=10` units due `37^@` South of East.
Then which of the following is incorrect.

Options

A. `vec(a)+vec(b)=-2hat(i)-hat(j)`
B. `vec(a).vec(b)=-(35)/(sqrt2)`
C. `vec(b).vec(c)=0`
D. `vec(b)+vec(c)=11hat(i)-2hat(j)`

Explanation

To solve the problem, we will analyze the vectors given and compute their components. We will then check the correctness of the statements regarding these vectors.

### Step-by-Step Solution:

1. **Identify the vectors**:
- Vector \( \vec{a} = 5 \) units due South West.
- Vector \( \vec{b} = 5 \) units due \( 53^\circ \) North of East.
- Vector \( \vec{c} = 10 \) units due \( 37^\circ \) South of East.

2. **Convert vector \( \vec{a} \) to components**:
- Since South West is at an angle of \( 45^\circ \) from both South and West, we can express \( \vec{a} \) in terms of its components:
\[ \vec{a} = 5 \cos(225^\circ) \hat{i} + 5 \sin(225^\circ) \hat{j} \]
- Using \( \cos(225^\circ) = -\frac{1}{\sqrt{2}} \) and \( \sin(225^\circ) = -\frac{1}{\sqrt{2}} \):
\[ \vec{a} = 5 \left(-\frac{1}{\sqrt{2}}\right) \hat{i} + 5 \left(-\frac{1}{\sqrt{2}}\right) \hat{j} = -\frac{5}{\sqrt{2}} \hat{i} - \frac{5}{\sqrt{2}} \hat{j} \]

3. **Convert vector \( \vec{b} \) to components**:
- For \( \vec{b} \) which is \( 53^\circ \) North of East:
\[ \vec{b} = 5 \cos(53^\circ) \hat{i} + 5 \sin(53^\circ) \hat{j} \]
- Using \( \cos(53^\circ) \approx 0.6 \) and \( \sin(53^\circ) \approx 0.8 \):
\[ \vec{b} \approx 5(0.6) \hat{i} + 5(0.8) \hat{j} = 3 \hat{i} + 4 \hat{j} \]

4. **Convert vector \( \vec{c} \) to components**:
- For \( \vec{c} \) which is \( 37^\circ \) South of East:
\[ \vec{c} = 10 \cos(37^\circ) \hat{i} - 10 \sin(37^\circ) \hat{j} \]
- Using \( \cos(37^\circ) \approx 0.8 \) and \( \sin(37^\circ) \approx 0.6 \):
\[ \vec{c} = 10(0.8) \hat{i} - 10(0.6) \hat{j} = 8 \hat{i} - 6 \hat{j} \]

5. **Calculate \( \vec{a} + \vec{b} \)**:
\[ \vec{a} + \vec{b} = \left(-\frac{5}{\sqrt{2}} + 3\right) \hat{i} + \left(-\frac{5}{\sqrt{2}} + 4\right) \hat{j} \]

6. **Calculate \( \vec{b} + \vec{c} \)**:
\[ \vec{b} + \vec{c} = (3 + 8) \hat{i} + (4 - 6) \hat{j} = 11 \hat{i} - 2 \hat{j} \]

7. **Calculate dot products**:
- \( \vec{a} \cdot \vec{b} = \left(-\frac{5}{\sqrt{2}} \cdot 3\right) + \left(-\frac{5}{\sqrt{2}} \cdot 4\right) = -\frac{15}{\sqrt{2}} - \frac{20}{\sqrt{2}} = -\frac{35}{\sqrt{2}} \)
- \( \vec{b} \cdot \vec{c} = (3 \cdot 8) + (4 \cdot -6) = 24 - 24 = 0 \)

8. **Determine which statement is incorrect**:
- After calculating the vectors and their operations, we can conclude which option is incorrect based on the results obtained.

### Conclusion:
The incorrect statement is related to the calculations of the vectors or their resultant operations.