Let's solve each part step-by-step: ### Given: - Regular octagon: \(ABCDEFGH\) - \(\vec{AB} = \vec{u}\) - \(\vec{BC} = \vec{v}\) --- ## **a) Find \(\vec{GE}\) in terms of \(\vec{u}\) and \(\vec{v}\):** Let's trace the path from \(G\) to \(E\): - G → F (\(\vec{GF}\)) - F → E (\(\vec{FE}\)) In a regular octagon, all sides are equal, and moving from one vertex to the next is analogous to moving from \(A\) to \(B\) (\(\vec{u}\)), \(B\) to \(C\) (\(\vec{v}\)), etc., but depending on the direction. Let’s write: - \(\vec{GF}\) is the same direction as \(\vec{AB}\) (since both are one side forward in the octagon, but starting from different points): **\(\vec{GF} = \vec{u}\)** - \(\vec{FE}\) is the same as \(\vec{BC} = \vec{v}\) So: \[ \vec{GE} = \vec{GF} + \vec{FE} = \vec{u} + \vec{v} \] ### **Final Answer for a):** \[ \boxed{\vec{GE} = \vec{u} + \vec{v}} \] --- ## **b) Express \(\vec{CD}\) in terms of \(\vec{u}\) and \(\vec{v}\):** \(\vec{CD}\) is the vector from \(C\) to \(D\). Let’s analyze the direction: - \(\vec{AB}\) goes from \(A\) to \(B\) - \(\vec{BC}\) goes from \(B\) to \(C\) - \(\vec{CD}\) goes from \(C\) to \(D\) In a regular octagon, each side is related by a rotation of \(45^\circ\). If \(\vec{AB} = \vec{u}\) and \(\vec{BC} = \vec{v}\), then \(\vec{CD}\) is the next side, which should have the same relationship as the others. But we want to express \(\vec{CD}\) **in terms of** \(\vec{u}\) and \(\vec{v}\). Notice the pattern: - \(\vec{AB} = \vec{u}\) - \(\vec{BC} = \vec{v}\) - \(\vec{CD}\) is the next edge, and in a regular octagon, moving from one edge to the next is a rotation by \(45^\circ\). Since \(\vec{u}\) and \(\vec{v}\) are consecutive edge vectors, and each edge is a \(45^\circ\) rotation from the previous, the third edge \(\vec{CD}\) can be written as: \[ \vec{CD} = \vec{v} - \vec{u} \] Here’s why: - Think of the symmetry: \(\vec{u}\) is from \(A\) to \(B\), \(\vec{v}\) from \(B\) to \(C\). - The vector \(\vec{CD}\) is the same as the change from \(\vec{u}\) to \(\vec{v}\), or: move by \(\vec{v}\) then subtract the previous direction \(\vec{u}\). ### **Final Answer for b):** \[ \boxed{\vec{CD} = \vec{v} - \vec{u}} \] --- ## **Summary** **a) \(\vec{GE} = \vec{u} + \vec{v}\)** **b) \(\vec{CD} = \vec{v} - \vec{u}\)** Let me know if you need a geometric explanation or vector diagram!