Certainly! Let’s break down the solution into **two steps** with clear explanations and then provide the **final answer**. --- ## **Step 1: Express Forces in Cartesian (Vector) Form** ### **F1 (100 lb)** Given: - Acts in the \( YZ \)-plane. - Angle with Z-axis = \( 34^\circ \). - Magnitude = \( 100 \) lb. **Decomposition:** \[ \begin{align*} F_{1x} &= \\ F_{1y} &= 100 \sin(34^\circ) \approx 55.94 \\ F_{1z} &= 100 \cos(34^\circ) \approx 82.85 \end{align*} \] So, \[ \vec{F_1} = \hat{i} + 55.94 \hat{j} + 82.85 \hat{k} \] **Explanation:** We resolved \( F_1 \) into its Y and Z components using sine and cosine since it lies in the YZ-plane. --- ### **F2 (120 lb)** Given: - Acts in the XY-plane. - Angle with X-axis = \( 63^\circ \). - Magnitude = \( 120 \) lb. **Decomposition:** \[ \begin{align*} F_{2x} &= 120 \cos(63^\circ) \approx 54.48 \\ F_{2y} &= 120 \sin(63^\circ) \approx 106.83 \\ F_{2z} &= \end{align*} \] So, \[ \vec{F_2} = 54.48 \hat{i} + 106.83 \hat{j} + \hat{k} \] **Explanation:** We resolved \( F_2 \) into its X and Y components using cosine and sine since it lies in the XY-plane. --- ## **Step 2: Find Resultant Force and its Magnitude** ### **Add Components:** \[ \vec{R} = ( + 54.48) \hat{i} + (55.94 + 106.83) \hat{j} + (82.85 + ) \hat{k} \] \[ \vec{R} = 54.48 \hat{i} + 162.77 \hat{j} + 82.85 \hat{k} \] ### **Magnitude of Resultant Force:** \[ |\vec{R}| = \sqrt{(54.48)^2 + (162.77)^2 + (82.85)^2} \] \[ |\vec{R}| \approx \sqrt{2968.7 + 26500.1 + 6863.1} \approx \sqrt{36331.9} \approx 190.6 \text{ lb} \] **Explanation:** We summed the corresponding components of both forces to get the resultant, then used the Pythagorean theorem in 3D to find the magnitude. --- ## **Final Answer** - **Resultant Force in Cartesian Form:** \[ \boxed{54.48 \hat{i} + 162.77 \hat{j} + 82.85 \hat{k}\ \text{lb}} \] - **Magnitude of Resultant Force:** \[ \boxed{190.6\ \text{lb}} \] Let me know if you need the coordinate **direction angles** as well!