1. ABCDEF is a regular hexagon in which \( \overrightarrow{B C} \) rep- resents \( \underline{b} \) and \( \overrightarrow{F C} \) represents \( 2 \underline{a} \). Express the vectors \( \overrightarrow{A B}, \overrightarrow{C D} \) and \( \overrightarrow{B E} \) in terms of \( \underline{a} \) and \( \underline{b} \).

To express the vectors \( \overrightarrow{A B}, \overrightarrow{C D} \), and \( \overrightarrow{B E} \) in terms of \( \underline{a} \) and \( \underline{b} \), we first need to understand the geometry of the regular hexagon ABCDEF. 1. **Understanding the Hexagon:** - A regular hexagon can be inscribed in a circle, and each vertex can be represented in terms of complex numbers or vectors in a coordinate system. - The vertices can be labeled as follows: - \( A \) at \( (1, 0) \) - \( B \) at \( \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right) \) - \( C \) at \( \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right) \) - \( D \) at \( (-1, 0) \) - \( E \) at \( \left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right) \) - \( F \) at \( \left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right) \) 2. **Given Vectors:** - \( \overrightarrow{B C} = \underline{b} \) - \( \overrightarrow{F C} = 2 \underline{a} \) 3. **Calculating the Vectors:** - The vector \( \overrightarrow{B C} \) can be calculated as: \[ \overrightarrow{B C} = \overrightarrow{C} - \overrightarrow{B} = \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right) - \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right) = (-1, 0) \] - Therefore, we have: \[ \underline{b} = (-1, 0) \] - The vector \( \overrightarrow{F C} \) can be calculated as: \[ \overrightarrow{F C} = \overrightarrow{C} - \overrightarrow{F} = \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right) - \left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right) = (-1, \sqrt{3}) \] - Therefore, we have: \[ 2 \underline{a} = (-1, \sqrt{3}) \implies \underline{a} = \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right) \] 4. **Expressing Other Vectors:** - Now we can express the required vectors: - **For \( \overrightarrow{A B} \):** \[ \overrightarrow{A B} = \overrightarrow{B} - \overrightarrow{A} = \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right) - (1, 0) = \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right) = \underline{a} \] - **For \( \overrightarrow{C D} \):** \[ \overrightarrow{C D} = \overrightarrow{D} - \overrightarrow{C} = (-1, 0) - \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right) = \left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right) = -\underline{a} \] - **For \( \overrightarrow{B E} \):** \[ \overrightarrow{B E} = \overrightarrow{E} - \overrightarrow{B} = \left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right) - \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right) = (-1, -\sqrt{3}) = -2\underline{a} \] 5. **Final Expressions:** - Thus, we have: - \( \overrightarrow{A B} = \underline{a} \) - \( \overrightarrow{C D} = -\underline{a} \) - \( \overrightarrow{B E} = -2\underline{a} \) In summary: - \( \overrightarrow{A B} = \underline{a} \) - \( \overrightarrow{C D} = -\underline{a} \) - \( \overrightarrow{B E} = -2\underline{a} \)