What is the value of `(vec(A) + vec(B)) .(vec(A) xx vec(B))` ?

Options

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

Answer

To solve the question, we need to find the value of the expression \( (\vec{A} + \vec{B}) \cdot (\vec{A} \times \vec{B}) \).

### Step-by-Step Solution:

1. **Understanding the Vectors**:
- Let \(\vec{A}\) and \(\vec{B}\) be two vectors in a plane. The vector \(\vec{A} + \vec{B}\) represents the resultant vector when both vectors are added together.

2. **Cross Product**:
- The cross product \(\vec{A} \times \vec{B}\) gives a vector that is perpendicular to the plane formed by \(\vec{A}\) and \(\vec{B}\). This means that \(\vec{A} \times \vec{B}\) is orthogonal to both \(\vec{A}\) and \(\vec{B}\).

3. **Perpendicularity**:
- Since \(\vec{A} + \vec{B}\) lies in the same plane as \(\vec{A}\) and \(\vec{B}\), and \(\vec{A} \times \vec{B}\) is perpendicular to this plane, it follows that \(\vec{A} + \vec{B}\) is also perpendicular to \(\vec{A} \times \vec{B}\).

4. **Dot Product of Perpendicular Vectors**:
- The dot product of two vectors that are perpendicular to each other is zero. Therefore, we can conclude that:
\[ (\vec{A} + \vec{B}) \cdot (\vec{A} \times \vec{B}) = 0 \]

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