If the vectors `vec(k) and vec(A)` are parallel to each other, then what is `k vec(k) xx vec(A)` equal to?

Options

A. `k^(2) vec(A)`
B. `vec(0)`
C. `-k^(2) vec(A)`
D. `vec(A)`

Answer

To solve the problem, we need to analyze the relationship between the vectors \( \vec{k} \) and \( \vec{A} \) being parallel and how it affects the cross product \( k \vec{k} \times \vec{A} \).

### Step-by-Step Solution:

1. **Understanding Parallel Vectors**:
Since the vectors \( \vec{k} \) and \( \vec{A} \) are parallel, we can express one vector as a scalar multiple of the other. Thus, we can write:
\[ \vec{k} = \lambda \vec{A} \]
where \( \lambda \) is some scalar.

2. **Cross Product Definition**:
The cross product of two vectors \( \vec{u} \) and \( \vec{v} \) is given by:
\[ \vec{u} \times \vec{v} = |\vec{u}| |\vec{v}| \sin(\theta) \hat{n} \]
where \( \theta \) is the angle between the two vectors and \( \hat{n} \) is the unit vector perpendicular to the plane formed by \( \vec{u} \) and \( \vec{v} \).

3. **Angle Between Parallel Vectors**:
For parallel vectors, the angle \( \theta \) between them is either \( 0^\circ \) or \( 180^\circ \). In both cases, \( \sin(\theta) = 0 \).

4. **Calculating the Cross Product**:
Therefore, when we compute \( \vec{k} \times \vec{A} \):
\[ \vec{k} \times \vec{A} = |\vec{k}| |\vec{A}| \sin(0) \hat{n} = 0 \]
This means that the cross product \( \vec{k} \times \vec{A} \) results in the zero vector.

5. **Final Calculation**:
Now, we need to find \( k \vec{k} \times \vec{A} \):
\[ k (\vec{k} \times \vec{A}) = k \cdot 0 = 0 \]

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