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

Answer

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: Analyze the Cross Product
The cross product \(\vec{B} \times \vec{A}\) produces 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: Determine the Dot Product
Since \(\vec{A}\) is one of the vectors used to form the cross product \(\vec{B} \times \vec{A}\), the resulting vector \(\vec{B} \times \vec{A}\) is perpendicular to \(\vec{A}\). Therefore, the dot product of \(\vec{A}\) with any vector that is perpendicular to it will be zero.

### Step 4: Conclude the Result
Thus, we have:
\[ \vec{A} \cdot (\vec{B} \times \vec{A}) = 0 \]

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

---