The angle between the vector `vec(A)` and `vec(B)` is `theta`. Find the value of triple product `vec(A).(vec(B)xxvec(A))`.

Options

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

Explanation

To solve the problem of finding the value of the triple product \( \vec{A} \cdot (\vec{B} \times \vec{A}) \), we can follow these steps:

### Step 1: Understand the Triple Product
The expression \( \vec{A} \cdot (\vec{B} \times \vec{A}) \) represents the dot product of vector \( \vec{A} \) with the cross product of vectors \( \vec{B} \) and \( \vec{A} \).

### Step 2: Properties of the Cross Product
The cross product \( \vec{B} \times \vec{A} \) results in a vector that is perpendicular to both \( \vec{B} \) and \( \vec{A} \). This means that the direction of \( \vec{B} \times \vec{A} \) is orthogonal to the plane formed by \( \vec{A} \) and \( \vec{B} \).

### Step 3: Dot Product with Perpendicular Vector
Since \( \vec{A} \) and \( \vec{B} \times \vec{A} \) are perpendicular to each other, the dot product \( \vec{A} \cdot (\vec{B} \times \vec{A}) \) will be zero. This is because the dot product of two perpendicular vectors is always zero.

### Step 4: Conclusion
Thus, we conclude that:
\[ \vec{A} \cdot (\vec{B} \times \vec{A}) = 0 \]

### Final Answer
The value of the triple product \( \vec{A} \cdot (\vec{B} \times \vec{A}) \) is \( 0 \).

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