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)sintheta`
D. `A^(2)Bcos theta`

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 triple product \( \vec{A} \cdot (\vec{B} \times \vec{A}) \) involves a dot product of vector \( \vec{A} \) with the cross product of vectors \( \vec{B} \) and \( \vec{A} \).

### Step 2: Analyze the Cross Product
The result of the cross product \( \vec{B} \times \vec{A} \) is a vector that is perpendicular to both \( \vec{B} \) and \( \vec{A} \). Therefore, the angle between \( \vec{A} \) and \( \vec{B} \times \vec{A} \) is \( 90^\circ \).

### Step 3: Apply the Dot Product
The dot product of two vectors is given by:
\[ \vec{X} \cdot \vec{Y} = |\vec{X}| |\vec{Y}| \cos(\phi) \]
where \( \phi \) is the angle between the two vectors. In our case, since \( \vec{A} \) is perpendicular to \( \vec{B} \times \vec{A} \), we have \( \phi = 90^\circ \).

### Step 4: Calculate the Dot Product
Since the cosine of \( 90^\circ \) is zero:
\[ \vec{A} \cdot (\vec{B} \times \vec{A}) = |\vec{A}| |\vec{B} \times \vec{A}| \cos(90^\circ) = 0 \]

### Conclusion
Thus, the value of the triple product \( \vec{A} \cdot (\vec{B} \times \vec{A}) \) is:
\[ \boxed{0} \]

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