A vector is not changed if

Options

A. it is rotated through an arbitrary angle
B. it is multiplied by an arbitrary scalar
C. it is cross multiplied by a unit vector
D. it is displaced paralled to itself

Answer

To solve the question "A vector is not changed if," we need to analyze each of the given options and determine which one does not alter the vector.

**Step 1: Understand the nature of vectors**
A vector is defined by both its magnitude and direction. Any operation that changes either of these two characteristics will change the vector itself.

**Step 2: Analyze Option 1 - Rotated through an arbitrary angle**
When a vector is rotated through an arbitrary angle, its direction changes. Therefore, the vector is not the same after rotation.
- **Conclusion:** Option 1 is incorrect.

**Step 3: Analyze Option 2 - Multiplied by an arbitrary scalar**
Multiplying a vector by a scalar changes its magnitude. Even if the direction remains the same, the vector's length changes, which means it is not the same vector.
- **Conclusion:** Option 2 is incorrect.

**Step 4: Analyze Option 3 - Cross multiplied by a unit vector**
Cross multiplying a vector by another vector (including a unit vector) results in a new vector that is perpendicular to both original vectors. This operation changes the direction and magnitude of the original vector.
- **Conclusion:** Option 3 is incorrect.

**Step 5: Analyze Option 4 - Displaced parallel to itself**
Displacing a vector parallel to itself means moving it in the same direction without changing its magnitude or direction. This operation does not alter the vector, as it remains the same in terms of both magnitude and direction.
- **Conclusion:** Option 4 is correct.

**Final Answer:**
A vector is not changed if it is displaced parallel to itself (Option 4).

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