Angular displacement is a vector quantity for __________ angles. (very small/even large)

Answer

To determine whether angular displacement is a vector quantity for very small angles or very large angles, we can analyze the properties of vector quantities and how they behave under addition.

### Step-by-Step Solution:

1. **Understanding Angular Displacement**:
- Angular displacement is defined as the angle through which an object has rotated about a specific axis in a given time period. It is represented by the symbol \( \Delta \theta \).

2. **Vector Quantity Characteristics**:
- A vector quantity has both magnitude and direction. Examples include displacement, velocity, and force. For a quantity to be classified as a vector, it must obey the rules of vector addition, such as the triangle law of vector addition.

3. **Small vs. Large Angles**:
- For small angles, the angular displacement can be treated as a vector. When two small angular displacements are added, they can be represented graphically as vectors, and the resultant can be found using vector addition rules. This is because the angle between the vectors remains small, and the sine of the angle between them can be approximated as the angle itself in radians.

4. **Graphical Representation**:
- When we draw vectors for small angles, they can be placed head-to-tail, and the resultant can be easily determined. This graphical addition is valid because the small angles do not significantly deviate from linearity.

5. **Conclusion**:
- For very large angles, the situation changes. As the angle increases, the direction of the angular displacement becomes less straightforward, and the addition of such large angles does not follow the simple vector addition rules. Therefore, the angular displacement is not treated as a vector quantity for large angles.

Thus, we conclude that **angular displacement is a vector quantity for very small angles**.

### Final Answer:
Angular displacement is a vector quantity for **very small** angles.

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