A

Step 1: Understand the definition of a vector. A vector has both magnitude and direction. Changes in these can affect the vector's representation.
Step 2: Analyze Option A: Displacing a vector parallel to itself means that its direction and magnitude remain unchanged. Thus, the vector itself remains the same in the context of vector representation.
Step 3: Review Option B: Rotating a vector through an arbitrary angle changes its direction, hence the vector is altered.
Step 4: Consider Option C: Cross-multiplying a vector by a unit vector results in a new vector that is perpendicular to both vectors. Therefore, this changes the original vector.
Step 5: Look at Option D: Multiplying a vector by an arbitrary scalar changes its magnitude (though the direction may remain the same if the scalar is positive).
Conclusion: The only option that leaves the vector unchanged is Option A. Therefore, the correct answer is A.