1. **Problem Statement:** Given vectors Q = 200 lb at 120°, P = 722 lb at 60°, F = 448 lb at 210°, and T = 400 lb at 290°, find: a. The x and y components of each vector. b. The components of each vector in a rotated coordinate system x'-y' inclined 10° counterclockwise from the original x-y axes. --- 2. **Formulas and Rules:** - To find components of a vector $V$ with magnitude $|V|$ and angle $\theta$ (measured counterclockwise from positive x-axis): $$V_x = |V| \cos \theta$$ $$V_y = |V| \sin \theta$$ - To find components in rotated axes (rotated by angle $\alpha$): $$V_{x'} = V_x \cos \alpha + V_y \sin \alpha$$ $$V_{y'} = -V_x \sin \alpha + V_y \cos \alpha$$ - Angles must be in degrees and converted to radians if using a calculator that requires it. --- 3. **Step a: Calculate x and y components** - For Q (200 lb, 120°): $$Q_x = 200 \cos 120^\circ = 200 \times (-0.5) = -100$$ $$Q_y = 200 \sin 120^\circ = 200 \times 0.8660 = 173.2$$ - For P (722 lb, 60°): $$P_x = 722 \cos 60^\circ = 722 \times 0.5 = 361$$ $$P_y = 722 \sin 60^\circ = 722 \times 0.8660 = 625.3$$ - For F (448 lb, 210°): $$F_x = 448 \cos 210^\circ = 448 \times (-0.8660) = -388.1$$ $$F_y = 448 \sin 210^\circ = 448 \times (-0.5) = -224$$ - For T (400 lb, 290°): $$T_x = 400 \cos 290^\circ = 400 \times 0.3420 = 136.8$$ $$T_y = 400 \sin 290^\circ = 400 \times (-0.9397) = -375.9$$ --- 4. **Step b: Calculate components in rotated axes (x', y') with $\alpha = 10^\circ$** - Compute $\cos 10^\circ = 0.9848$, $\sin 10^\circ = 0.1736$ - For Q: $$Q_{x'} = (-100)(0.9848) + (173.2)(0.1736) = -98.48 + 30.06 = -68.42$$ $$Q_{y'} = -(-100)(0.1736) + (173.2)(0.9848) = 17.36 + 170.56 = 187.92$$ - For P: $$P_{x'} = 361(0.9848) + 625.3(0.1736) = 355.6 + 108.5 = 464.1$$ $$P_{y'} = -361(0.1736) + 625.3(0.9848) = -62.7 + 615.9 = 553.2$$ - For F: $$F_{x'} = (-388.1)(0.9848) + (-224)(0.1736) = -382.1 - 38.9 = -421.0$$ $$F_{y'} = -(-388.1)(0.1736) + (-224)(0.9848) = 67.4 - 220.6 = -153.2$$ - For T: $$T_{x'} = 136.8(0.9848) + (-375.9)(0.1736) = 134.7 - 65.3 = 69.4$$ $$T_{y'} = -136.8(0.1736) + (-375.9)(0.9848) = -23.7 - 370.3 = -394.0$$ --- **Final answers:** | Vector | $V_x$ | $V_y$ | $V_{x'}$ | $V_{y'}$ | |--------|-------|-------|----------|----------| | Q | -100 | 173.2 | -68.42 | 187.92 | | P | 361 | 625.3 | 464.1 | 553.2 | | F | -388.1| -224 | -421.0 | -153.2 | | T | 136.8 | -375.9| 69.4 | -394.0 |