1. **Problem:** A courier company has a central depot at point O. A delivery motorbike follows the route: Depot (O) → Address A: $\vec{v}_1 = 8i + 10j$ km; A → B: $\vec{v}_2 = 7i + 14j$ km; B → C: $\vec{v}_3 = 10i - 8j$ km; C → Post Office: $\vec{v}_4 = 5i + 6j$ km. Find: 2. **Step 1: Find the final position vector of the motorbike relative to depot using polygon rule.** The polygon rule states that the resultant vector is the vector sum of all displacement vectors: $$\vec{R} = \vec{v}_1 + \vec{v}_2 + \vec{v}_3 + \vec{v}_4$$ Calculate each component: $$R_x = 8 + 7 + 10 + 5 = 30$$ $$R_y = 10 + 14 - 8 + 6 = 22$$ So, $$\vec{R} = 30i + 22j$$ 3. **Step 2: Calculate the straight-line distance from depot to Post Office.** Distance is magnitude of $\vec{R}$: $$d = \sqrt{30^2 + 22^2} = \sqrt{900 + 484} = \sqrt{1384} \approx 37.22\text{ km}$$ 4. **Step 3: Calculate total distance traveled following the route.** Sum of magnitudes of each vector: $$|\vec{v}_1| = \sqrt{8^2 + 10^2} = \sqrt{64 + 100} = \sqrt{164} \approx 12.81$$ $$|\vec{v}_2| = \sqrt{7^2 + 14^2} = \sqrt{49 + 196} = \sqrt{245} \approx 15.65$$ $$|\vec{v}_3| = \sqrt{10^2 + (-8)^2} = \sqrt{100 + 64} = \sqrt{164} \approx 12.81$$ $$|\vec{v}_4| = \sqrt{5^2 + 6^2} = \sqrt{25 + 36} = \sqrt{61} \approx 7.81$$ Total distance: $$D = 12.81 + 15.65 + 12.81 + 7.81 = 49.08\text{ km}$$ 5. **Step 4: If the motorbike returns directly to depot from Post Office:** - Displacement vector needed is the negative of $\vec{R}$: $$\vec{v}_{return} = -\vec{R} = -30i - 22j$$ - Distance to travel back: $$d_{return} = |\vec{v}_{return}| = |\vec{R}| = 37.22\text{ km}$$ 6. **Step 5: Compare efficiency of the route:** - Total route distance: $49.08$ km - Direct round trip distance: $2 \times 37.22 = 74.44$ km - Difference: $$\Delta = 74.44 - 49.08 = 25.36\text{ km}$$ - Percentage difference relative to direct round trip: $$\%\text{ difference} = \frac{25.36}{74.44} \times 100\% \approx 34.08\%$$ **Answer:** a. Final position vector relative to depot: $\boxed{30i + 22j}$ km b. Straight-line distance depot to Post Office: $\boxed{37.22}$ km c. Total distance traveled: $\boxed{49.08}$ km d. Return vector: $\boxed{-30i - 22j}$ km, return distance: $\boxed{37.22}$ km e. Route efficiency: direct round trip is $74.44$ km, route traveled is $49.08$ km, saving about $\boxed{34.08\%}$ in distance.