Let's break down the problem and solve step-by-step: ## **Given Vectors** 1. **Vector A:** \( 357\, \text{m} \) East - \( x_1 = 357\, \text{m} \), \( y_1 = \) 2. **Vector B:** \( 764\, \text{m} \) at \( 25^\circ \) North of East - \( x_2 = 764 \cos 25^\circ = 764 \times .9063 = 692.4\, \text{m} \) - \( y_2 = 764 \sin 25^\circ = 764 \times .4226 = 322.8\, \text{m} \) 3. **Vector C:** \( 465\, \text{m} \) Southwest (Southwest = \( 225^\circ \) from East, or \( 45^\circ \) South of West, so both components are negative) - \( x_3 = -465 \cos 45^\circ = -465 \times .7071 = -328.8\, \text{m} \) - \( y_3 = -465 \sin 45^\circ = -465 \times .7071 = -328.8\, \text{m} \) --- ## **Step 1: Add Components** ### **Sum of \( x \)-components:** \[ R_x = x_1 + x_2 + x_3 = 357 + 692.4 - 328.8 = 720.6\, \text{m} \] ### **Sum of \( y \)-components:** \[ R_y = y_1 + y_2 + y_3 = + 322.8 - 328.8 = -6.\, \text{m} \] --- ## **Step 2: Write in Unit Vector Notation** \[ \vec{R} = (720.6\, \hat{i} - 6.\, \hat{j})\, \text{m} \] --- ## **Step 3: Find Magnitude of Resultant** \[ |\vec{R}| = \sqrt{(720.6)^2 + (-6.)^2} \] \[ |\vec{R}| = \sqrt{519264.36 + 36} = \sqrt{519300.36} \approx 720.6\, \text{m} \] --- ## **Step 4: Find Direction (Angle θ with Respect to East)** \[ \theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) \] \[ \theta = \tan^{-1}\left(\frac{-6.}{720.6}\right) \approx \tan^{-1}(-.00833) \approx -.477^{\circ} \] **This is almost due East, but slightly South of East.** --- ## **Final Answers** ### (a) **Unit Vector Notation:** \[ \boxed{ \vec{R} = 720.6\, \hat{i} - 6.\, \hat{j}~\text{m} } \] ### (b) **Magnitude and Displacement:** \[ \boxed{ |\vec{R}| = 720.6~\text{m},~.48^\circ~\text{South of East} } \] --- **Summary Table:** | Vector | \( x \) (m) | \( y \) (m) | |--------|-------------|-------------| | A | 357 | | | B | 692.4 | 322.8 | | C | -328.8 | -328.8 | | **Sum**| **720.6** | **-6.** |