1. **State the problem:** Find the resultant vector of two vectors given in polar form: $3.51\angle 21.8^\circ$ and $6.24\angle 51.5^\circ$. 2. **Formula and rules:** To add vectors in polar form, convert each vector to rectangular coordinates using: $$x = r \cos \theta$$ $$y = r \sin \theta$$ where $r$ is the magnitude and $\theta$ is the angle. 3. **Convert first vector:** $$x_1 = 3.51 \cos 21.8^\circ = 3.51 \times 0.928 = 3.256$$ $$y_1 = 3.51 \sin 21.8^\circ = 3.51 \times 0.371 = 1.303$$ 4. **Convert second vector:** $$x_2 = 6.24 \cos 51.5^\circ = 6.24 \times 0.622 = 3.882$$ $$y_2 = 6.24 \sin 51.5^\circ = 6.24 \times 0.783 = 4.886$$ 5. **Add components:** $$x_R = x_1 + x_2 = 3.256 + 3.882 = 7.138$$ $$y_R = y_1 + y_2 = 1.303 + 4.886 = 6.189$$ 6. **Find magnitude of resultant:** $$R = \sqrt{x_R^2 + y_R^2} = \sqrt{7.138^2 + 6.189^2} = \sqrt{50.96 + 38.30} = \sqrt{89.26} = 9.45$$ 7. **Find angle of resultant:** $$\theta_R = \tan^{-1} \left( \frac{y_R}{x_R} \right) = \tan^{-1} \left( \frac{6.189}{7.138} \right) = \tan^{-1}(0.867) = 40.9^\circ$$ **Final answer:** The resultant vector is $9.45 \angle 40.9^\circ$.