Let's break down the matching for the vector answers related to the equilibrium problem. ### Problem Overview - You have a 200-kg plate supported by hinges A and B. - A cable DCE passes over a frictionless hook at C. - The tension is the same in both parts of the cable. - You are to determine: - The tension in the cable (TCD and TCE) - The reactions at A (FA) and B (FB) ### Given Vector Answers 1. (211.601, 708.5, 312.2) 2. (-442.421, 442.421, -316.015) 3. (230.82, 538.58, -384.7) 4. (0, 272.5, 388.515) ### Matching We need to match the vectors to the physical quantities: #### 1. TCD (Tension in cable from D to C) #### 2. TCE (Tension in cable from C to E) #### 3. FA (Reaction at hinge A) #### 4. FB (Reaction at hinge B) #### **Step-by-step Matching:** - **Tensions in the cable (TCD, TCE):** These should be the same magnitude but opposite in direction, since the cable passes over a frictionless pulley at C. - **Reactions at hinges (FA, FB):** These are typically support reactions and might sum to the weight of the plate in the vertical direction. #### **Vector Analysis:** - The only pair of vectors that are exact negatives of each other in all three components are: - (211.601, 708.5, 312.2) and (-442.421, 442.421, -316.015) are not exact negatives, but check if any pair has this property. - (211.601, 708.5, 312.2) and (230.82, 538.58, -384.7) are not negatives. - (230.82, 538.58, -384.7) and (0, 272.5, 388.515) are not negatives. - (-442.421, 442.421, -316.015) and (0, 272.5, 388.515) are not negatives. Since none are exact negatives, let's look at the **direction** and **magnitude**: - **Reactions at hinges**: These are typically not parallel and can have both positive and negative components depending on the direction of force they counteract. - **Tensions in cable**: Should be along the direction of the cable segments. #### **Match Based on Convention**: - TCD and TCE: Should be the largest magnitude vectors, likely (211.601, 708.5, 312.2) and (-442.421, 442.421, -316.015). - FA and FB: Typically smaller, likely (230.82, 538.58, -384.7) and (0, 272.5, 388.515). ### **Final Matching:** 1. (211.601, 708.5, 312.2) → **TCD** 2. (-442.421, 442.421, -316.015) → **TCE** 3. (230.82, 538.58, -384.7) → **FA** 4. (0, 272.5, 388.515) → **FB** --- **Summary Table:** | Vector | Quantity | |-------------------------------|----------| | (211.601, 708.5, 312.2) | TCD | | (-442.421, 442.421, -316.015) | TCE | | (230.82, 538.58, -384.7) | FA | | (0, 272.5, 388.515) | FB | Let me know if you need a step-by-step calculation or further explanation!