Let’s break down the steps for finding the resultant vector: **Given:** - \( A = 15.0 \, \text{m} \) - \( B = 10.0 \, \text{m} \) - \( C = 13.0 \, \text{m} \) - \( D = 30.0 \, \text{m} \) Angles from the diagram (measured counterclockwise from the +x-axis): - \( A \): along +y-axis → \( 90^\circ \) - \( B \): \( 30^\circ \) below +x-axis → \( -30^\circ \) - \( C \): \( 30^\circ \) above -x-axis → \( 180^\circ - 30^\circ = 150^\circ \) - \( D \): \( 60^\circ \) below -x-axis → \( 180^\circ + 60^\circ = 240^\circ \) We will resolve each vector into x and y components. --- ## **Step 1: Resolve Each Vector into Components** **A:** - \( A_x = 15.0 \cos 90^\circ = 0 \) - \( A_y = 15.0 \sin 90^\circ = 15.0 \) **B:** - \( B_x = 10.0 \cos(-30^\circ) = 10.0 \times 0.866 = 8.66 \) - \( B_y = 10.0 \sin(-30^\circ) = 10.0 \times (-0.5) = -5.0 \) **C:** - \( C_x = 13.0 \cos(150^\circ) = 13.0 \times (-0.866) = -11.26 \) - \( C_y = 13.0 \sin(150^\circ) = 13.0 \times 0.5 = 6.5 \) **D:** - \( D_x = 30.0 \cos(240^\circ) = 30.0 \times (-0.5) = -15.0 \) - \( D_y = 30.0 \sin(240^\circ) = 30.0 \times (-0.866) = -25.98 \) --- ## **Step 2: Add All x and y Components** \[ R_x = A_x + B_x + C_x + D_x = 0 + 8.66 - 11.26 - 15.0 = -17.6 \] \[ R_y = A_y + B_y + C_y + D_y = 15.0 - 5.0 + 6.5 - 25.98 = -9.48 \] --- ## **Step 3: Find the Magnitude of the Resultant Vector** \[ R = \sqrt{R_x^2 + R_y^2} = \sqrt{(-17.6)^2 + (-9.48)^2} = \sqrt{309.76 + 89.86} = \sqrt{399.62} \approx 20.0 \] --- ## **Step 4: Find the Direction (Angle from the +x-axis, counterclockwise)** \[ \theta = \tan^{-1} \left(\frac{R_y}{R_x}\right) = \tan^{-1} \left(\frac{-9.48}{-17.6}\right) = \tan^{-1}(0.539) \] \[ \theta \approx 28.3^\circ \] Both \( R_x \) and \( R_y \) are negative, so the vector is in the **third quadrant**. To get the angle counterclockwise from the +x-axis: \[ \theta_{ccw} = 180^\circ + 28.3^\circ = 208.3^\circ \] --- ## **Final Answers:** ### (a) Magnitude: \[ \boxed{20.0 \ \text{m}} \] ### (b) Direction (counterclockwise from +x-axis): \[ \boxed{208^\circ} \] --- **Summary Table:** | Vector | \( x \) component | \( y \) component | |--------|------------------|-------------------| | A | 0 | 15.0 | | B | +8.66 | -5.0 | | C | -11.26 | +6.5 | | D | -15.0 | -25.98 | |**Total** | **-17.6** | **-9.48** | --- **Let me know if you need the detailed breakdown of any step!**