If `vec(A), vec(B)` and `vec(C )` are three vectors, then the wrong relation is :

Options

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

Answer

To solve the question, we need to analyze the given options and determine which one represents a wrong relation among the three vectors \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\).

### Step-by-Step Solution:

1. **Option 1: \(\vec{A} + \vec{B} + \vec{C} = \vec{A} + \vec{B} + \vec{C}\)**
- This statement is trivially true as both sides of the equation are identical. This follows from the property of equality.

**Hint:** Check if both sides of the equation are the same.

2. **Option 2: \(\vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}\)**
- This statement represents the distributive property of the dot product over vector addition. It is a valid relation in vector algebra.

**Hint:** Recall the distributive property of dot products.

3. **Option 3: \(\vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}\)**
- This is similar to Option 2 and also represents the distributive property of the dot product. Hence, this statement is correct.

**Hint:** Verify if the distributive property applies to the operation in question.

4. **Option 4: \((\vec{A} \times \vec{B}) \cdot \vec{C} = \vec{A} \times (\vec{B} \cdot \vec{C})\)**
- The left-hand side \((\vec{A} \times \vec{B}) \cdot \vec{C}\) represents the scalar product of the vector \(\vec{C}\) with the cross product of \(\vec{A}\) and \(\vec{B}\), which is a valid operation. However, the right-hand side \(\vec{A} \times (\vec{B} \cdot \vec{C})\) is meaningless because \(\vec{B} \cdot \vec{C}\) results in a scalar, and you cannot take the cross product of a vector with a scalar. Therefore, this statement is incorrect.

**Hint:** Identify the properties of vector operations, especially the distinction between scalar and vector products.

### Conclusion:
The wrong relation among the given options is **Option 4**.