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 solve the question, we need to analyze each of the given options and determine which one is incorrect. Let's go through each option step by step.

### Step-by-Step Solution:

1. **Analyze Option 1:**
- The statement is: \( \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{b} \cdot \mathbf{a} + \mathbf{c} \cdot \mathbf{a} \).
- This follows the distributive law of the dot product.
- **Conclusion:** This statement is correct.

**Hint:** Remember that the dot product is distributive over addition.

2. **Analyze Option 2:**
- The statement is: \( \mathbf{a} \times (\mathbf{b} + \mathbf{c}) = \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c} \).
- This follows the distributive law of the cross product.
- **Conclusion:** This statement is also correct.

**Hint:** The cross product also obeys the distributive law.

3. **Analyze Option 3:**
- The statement is: \( (\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{b} \cdot \mathbf{c}) = \mathbf{a} \times \mathbf{b} \cdot \mathbf{a} \times \mathbf{c} \).
- Here, \( \mathbf{b} \cdot \mathbf{c} \) is a scalar, and the left-hand side involves a scalar multiplying a vector, which is not a valid operation in vector algebra.
- **Conclusion:** This statement is incorrect.

**Hint:** Remember that a scalar cannot be crossed with a vector.

4. **Analyze Option 4:**
- The statement is: \( (\mathbf{b} \cdot \mathbf{c}) \mathbf{a} = \mathbf{a} (\mathbf{c} \cdot \mathbf{b}) \).
- This follows the commutative property of the dot product.
- **Conclusion:** This statement is correct.

**Hint:** The dot product is commutative, meaning \( \mathbf{b} \cdot \mathbf{c} = \mathbf{c} \cdot \mathbf{b} \).

### Final Conclusion:
The incorrect statement among the options provided is **Option 3**.