The problem involves rotating the vector $\overrightarrow{ON}$ about point $O$ by 120°.

Step-by-step analysis:

1. Understanding the setup:

   • The hexagon $GIKMPR$ is regular, meaning all sides and angles between adjacent sides are equal.
   • The dashed lines indicate that each angle between these lines is 30°.
   • $\overrightarrow{ON}$ refers to the vector from $O$ to point $N$.

2. Identifying the initial angle:

   • The line $ON$ forms an angle relative to a reference line, such as $OI$ or $OK$. Since this is a regular hexagon, each side is 60° apart. $N$ is two steps clockwise from $OI$, indicating that the angle $\overrightarrow{ON}$ makes with $OI$ is 60° clockwise.

3. Rotation by 120° about $O$:

   • A rotation of 120° counterclockwise (standard convention for positive rotation) moves the vector $ON$ two steps counterclockwise.
   • Starting from $N$, after rotating 120°, the image of $\overrightarrow{ON}$ lands at the position of $OL$, which is two positions counterclockwise from $ON$.

Conclusion:

The image of $\overrightarrow{ON}$ after a rotation of 120° about $O$ is $\overrightarrow{OL}$.

Thus, the correct answer is c. $\overrightarrow{OL}$.

Would you like further details or have any other questions? Here are five related questions you can consider:

1. What is the rule for rotating a point in a regular polygon?
2. How do we calculate the angle between adjacent vertices in a regular hexagon?
3. What happens to the coordinates of a point after a specific degree of rotation about the origin?
4. How does the concept of symmetry in regular polygons affect transformations like rotations?
5. What is the geometric relationship between the hexagon and the triangle in this figure?

Tip: When dealing with rotations, it’s useful to remember that counterclockwise is the standard positive direction for angles in most geometric contexts.