In Figure 1.24, 
𝐴
 , 
𝐵
 , 
𝐶
 , 
𝐷
 , 
𝐸
 , and 
𝐹
 are the vertices of a regular hexagon centered at the origin.

Express each of the following vectors in terms of 
 
𝑎
=
OA
→
 and 
 
𝑏
=
OB
→
 :

 
AB
→

 
BC
→

 
AD
→

 
CF
→

 


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!

---

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!

In Figure 1.24, A, B, C, D, E, and F are the vertices of a regular hexagon centered at the origin. Express each of the following vectors in terms of a = OA→ and b = OB→: AB→, BC→, AD→, CF→.

Solve vector expressions in a regular hexagon, centered at the origin. Learn how to express vectors such as AB→, BC→, AD→, and CF→ in terms of a = OA→ and b = OB→ using vector algebra. This problem involves concepts like vector subtraction, geometric symmetry, and rotation transformations, suitable for Grades 10-12.

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