The value of `(vec(A)+vec(B)).(vec(A)xxvec(B))` is :-

Options

A. 0
B. `A^(2)-B^(2)`
C. `A^(2)+B^(2)+2AB`
D. None of these

Answer

To solve the problem, we need to evaluate the expression \((\vec{A} + \vec{B}) \cdot (\vec{A} \times \vec{B})\).

### Step-by-Step Solution:

1. **Understand the Components**:
We have two vectors, \(\vec{A}\) and \(\vec{B}\). The expression involves both the dot product and the cross product of these vectors.

2. **Expand the Expression**:
We can use the distributive property of the dot product:
\[ (\vec{A} + \vec{B}) \cdot (\vec{A} \times \vec{B}) = \vec{A} \cdot (\vec{A} \times \vec{B}) + \vec{B} \cdot (\vec{A} \times \vec{B}) \]

3. **Evaluate Each Term**:
- The term \(\vec{A} \cdot (\vec{A} \times \vec{B})\): The cross product \(\vec{A} \times \vec{B}\) is a vector that is perpendicular to both \(\vec{A}\) and \(\vec{B}\). Therefore, the dot product of \(\vec{A}\) with a vector perpendicular to it is zero:
\[ \vec{A} \cdot (\vec{A} \times \vec{B}) = 0 \]
- The term \(\vec{B} \cdot (\vec{A} \times \vec{B})\): Similarly, the cross product \(\vec{A} \times \vec{B}\) is also perpendicular to \(\vec{B}\), hence:
\[ \vec{B} \cdot (\vec{A} \times \vec{B}) = 0 \]

4. **Combine the Results**:
Now, substituting back into our expression:
\[ (\vec{A} + \vec{B}) \cdot (\vec{A} \times \vec{B}) = 0 + 0 = 0 \]

5. **Final Answer**:
Thus, the value of \((\vec{A} + \vec{B}) \cdot (\vec{A} \times \vec{B})\) is:
\[ \boxed{0} \]