1. **Problem statement:** Given the vectors in the quadrilateral with points A, B, C, D, and the relations AB = DC, DA = CB, DE = EB, EA = CE, write each sum or difference as a single vector. 2. **Recall vector addition and subtraction rules:** - Vector addition is associative and commutative. - The sum of vectors along a path equals the vector from the start to the end point. - The difference of vectors can be interpreted as adding the negative of a vector. 3. **Solve each part:** **a.** $\mathbf{AB} + \mathbf{BC}$ - Vector $\mathbf{AB}$ goes from A to B. - Vector $\mathbf{BC}$ goes from B to C. - Adding these vectors corresponds to moving from A to B, then B to C, so the resultant vector is from A to C. $$\mathbf{AB} + \mathbf{BC} = \mathbf{AC}$$ **b.** $\mathbf{CD} + \mathbf{DB}$ - Vector $\mathbf{CD}$ goes from C to D. - Vector $\mathbf{DB}$ goes from D to B. - Adding these vectors corresponds to moving from C to D, then D to B, so the resultant vector is from C to B. $$\mathbf{CD} + \mathbf{DB} = \mathbf{CB}$$ **c.** $\mathbf{DB} - \mathbf{AB}$ - Vector $\mathbf{DB}$ goes from D to B. - Vector $\mathbf{AB}$ goes from A to B. - Subtracting $\mathbf{AB}$ is the same as adding $-\mathbf{AB}$, which goes from B to A. So, $$\mathbf{DB} - \mathbf{AB} = \mathbf{DB} + (-\mathbf{AB}) = \mathbf{DB} + \mathbf{BA}$$ - Vector $\mathbf{DB}$ goes from D to B, and $\mathbf{BA}$ goes from B to A, so the sum is from D to A. $$\mathbf{DB} - \mathbf{AB} = \mathbf{DA}$$ **d.** $\mathbf{DC} + \mathbf{CA} + \mathbf{AB}$ - Vector $\mathbf{DC}$ goes from D to C. - Vector $\mathbf{CA}$ goes from C to A. - Vector $\mathbf{AB}$ goes from A to B. Adding these vectors corresponds to moving from D to C, then C to A, then A to B, so the resultant vector is from D to B. $$\mathbf{DC} + \mathbf{CA} + \mathbf{AB} = \mathbf{DB}$$ 4. **Summary of results:** - a) $\mathbf{AB} + \mathbf{BC} = \mathbf{AC}$ - b) $\mathbf{CD} + \mathbf{DB} = \mathbf{CB}$ - c) $\mathbf{DB} - \mathbf{AB} = \mathbf{DA}$ - d) $\mathbf{DC} + \mathbf{CA} + \mathbf{AB} = \mathbf{DB}$ These results use the property that the sum of vectors along connected points equals the vector from the start to the end point.