We are given that $A, B, C, D, E, F$ are the vertices of a regular hexagon centered at the origin, with vectors $\mathbf{a} = \overrightarrow{OA}$ and $\mathbf{b} = \overrightarrow{OB}$.

Step 1: Understanding the Geometry of a Regular Hexagon

In a regular hexagon:

• All sides are equal.
• The angle between adjacent vertices at the origin is $60^\circ$.

If we take the origin as the center of the hexagon, we know that the vertices are located on the unit circle, with angles $0^\circ, 60^\circ, 120^\circ, 180^\circ, 240^\circ, 300^\circ$ (or equivalently, $\frac{\pi}{3}$ radians between each).

Let’s express the requested vectors in terms of $\mathbf{a}$ and $\mathbf{b}$.

Step 2: Expressing Vectors

1. $\overrightarrow{AB}$

We know that $\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA}$. Since $\mathbf{a} = \overrightarrow{OA}$ and $\mathbf{b} = \overrightarrow{OB}$, we can write: $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$

2. $\overrightarrow{BC}$

Next, $\overrightarrow{BC} = \overrightarrow{OC} - \overrightarrow{OB}$. If $\mathbf{c} = \overrightarrow{OC}$, then: $\overrightarrow{BC} = \mathbf{c} - \mathbf{b}$ We can express $\mathbf{c}$ in terms of $\mathbf{a}$ and $\mathbf{b}$ using geometry. Since the angle between $OA$ and $OB$ is $60^\circ$, and the angle between $OB$ and $OC$ is also $60^\circ$, $\mathbf{c}$ can be expressed as a rotation of $\mathbf{b}$ by $60^\circ$. Thus, we use: $\overrightarrow{BC} = \mathbf{a} - \mathbf{b}$

3. $\overrightarrow{AD}$

To calculate $\overrightarrow{AD} = \overrightarrow{OD} - \overrightarrow{OA}$, note that $D$ is directly opposite $A$, and hence, $\overrightarrow{OD} = -\mathbf{a}$. Therefore: $\overrightarrow{AD} = (-\mathbf{a}) - \mathbf{a} = -2\mathbf{a}$

4. $\overrightarrow{CF}$

Finally, $\overrightarrow{CF} = \overrightarrow{OF} - \overrightarrow{OC}$. We know that $F$ is one vertex counterclockwise from $A$, and thus $\overrightarrow{OF}$ is a rotation of $\mathbf{a}$ by $-60^\circ$. Similarly, $\overrightarrow{OC} = \mathbf{c}$, which we already know. Thus: $\overrightarrow{CF} = -\mathbf{a} - \mathbf{b}$

Summary of Results:

• $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$
• $\overrightarrow{BC} = \mathbf{a} - \mathbf{b}$
• $\overrightarrow{AD} = -2\mathbf{a}$
• $\overrightarrow{CF} = -\mathbf{a} - \mathbf{b}$

Let me know if you'd like any clarifications or further details!

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Here are 5 related questions you might explore:

1. How would you express the vector $\overrightarrow{AE}$ in terms of $\mathbf{a}$ and $\mathbf{b}$?
2. What is the geometric interpretation of the vector $\overrightarrow{AD}$ in the hexagon?
3. How would the vectors change if the hexagon were not centered at the origin?
4. Can you express the vectors using complex numbers instead of vectors $\mathbf{a}$ and $\mathbf{b}$?
5. How do the magnitudes of $\overrightarrow{AB}$ and $\overrightarrow{AD}$ compare?

Tip: For regular polygons, rotation transformations can simplify the process of finding relationships between vectors!