Which of the following is incorrect ?

Options

A. `vec(a).(vec(b) + vec(c )) = vec(b).vec(a) + vec(a).vec(c )`
B. `vec(a) xx (vec(b) + vec(c )) = (vec(a) xx vec(c )) + (vec(a) xx vec(b))`
C. `vec(a) xx (vec(b).vec(c )) = (vec(a) xx vec(b)).(vec(a) xx vec(c ))`
D. `(vec(b).vec(c ))vec(a) = vec(a)(vec(c ).vec(b))`

Answer

To determine which of the given options is incorrect, we will analyze each option based on the properties of vector operations, specifically focusing on the distributive property, dot product, and cross product.

### Step-by-Step Solution:

1. **Understanding Vector Operations**:
- Vectors can be added and multiplied using two primary operations: the dot product and the cross product.
- The dot product of two vectors results in a scalar, while the cross product results in another vector.

2. **Analyzing Each Option**:
- We will examine each option to see if it adheres to the properties of vector operations.

3. **Option A**:
- This option likely involves the distributive property of vector addition and multiplication.
- The distributive property states that \( \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c} \) and \( \mathbf{a} \times (\mathbf{b} + \mathbf{c}) = \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c} \).
- This is correct.

4. **Option B**:
- This option also likely involves the distributive property, specifically concerning the cross product.
- The expression \( \mathbf{a} \times (\mathbf{b} + \mathbf{c}) \) is valid and follows the distributive law.
- This is also correct.

5. **Option C**:
- This option states \( \mathbf{a} \times \mathbf{b} \cdot \mathbf{c} \).
- Here, we have a vector \( \mathbf{a} \times \mathbf{b} \) and a scalar \( \mathbf{c} \). The dot product between a vector and a scalar is not defined.
- Therefore, this expression is incorrect.

6. **Option D**:
- This option discusses the commutative property of the dot product, stating \( \mathbf{b} \cdot \mathbf{c} = \mathbf{c} \cdot \mathbf{b} \).
- This is true for the dot product, as it is commutative. The expression involving scalar multiplication is also valid.
- Thus, this option is correct.

7. **Conclusion**:
- The only incorrect option among the given choices is **Option C**.

### Final Answer:
**C is the incorrect option.**