If the angle between the vectors `vec A and vec B` is `theta`, the value of the product `(vec B xx vec A) cdot vec A` is equal to

Options

A. `BA^(2) sin theta`
B. `BA^(2) cos theta`
C. `BA^(2) sin theta cos theta`
D. zero

Explanation

To solve the problem, we need to find the value of the expression \((\vec{B} \times \vec{A}) \cdot \vec{A}\) given that the angle between the vectors \(\vec{A}\) and \(\vec{B}\) is \(\theta\).

### Step-by-Step Solution:

1. **Understand the Cross Product**:
The cross product \(\vec{B} \times \vec{A}\) results in a vector that is perpendicular to both \(\vec{B}\) and \(\vec{A}\). Let's denote this vector as \(\vec{C}\):
\[ \vec{C} = \vec{B} \times \vec{A} \]

2. **Properties of the Dot Product**:
The dot product of two vectors is defined as:
\[ \vec{X} \cdot \vec{Y} = |\vec{X}| |\vec{Y}| \cos(\phi) \]
where \(\phi\) is the angle between the two vectors \(\vec{X}\) and \(\vec{Y}\).

3. **Determine the Angle**:
Since \(\vec{C} = \vec{B} \times \vec{A}\) is perpendicular to both \(\vec{A}\) and \(\vec{B}\), the angle between \(\vec{C}\) and \(\vec{A}\) is \(90^\circ\).

4. **Calculate the Dot Product**:
Now, we can evaluate the expression \((\vec{B} \times \vec{A}) \cdot \vec{A}\):
\[ (\vec{B} \times \vec{A}) \cdot \vec{A} = |\vec{C}| |\vec{A}| \cos(90^\circ) \]
Since \(\cos(90^\circ) = 0\), we have:
\[ (\vec{B} \times \vec{A}) \cdot \vec{A} = |\vec{C}| |\vec{A}| \cdot 0 = 0 \]

5. **Conclusion**:
Therefore, the value of the product \((\vec{B} \times \vec{A}) \cdot \vec{A}\) is:
\[ \boxed{0} \]