1. **State the problem:** We are given two vectors \( \mathbf{u} = 4 \langle \cos 55^\circ, \sin 55^\circ \rangle \) and \( \mathbf{v} = 2 \langle \cos 350^\circ, \sin 350^\circ \rangle \). We need to find their sum \( \mathbf{u} + \mathbf{v} \) and express it in magnitude and direction form, rounding magnitude to the nearest tenth and direction to the nearest degree with \(0^\circ \leq \theta < 360^\circ\). 2. **Write the vectors in component form:** \[ \mathbf{u} = 4 \langle \cos 55^\circ, \sin 55^\circ \rangle = 4 \langle 0.5736, 0.8192 \rangle = \langle 2.2944, 3.2768 \rangle \] \[ \mathbf{v} = 2 \langle \cos 350^\circ, \sin 350^\circ \rangle = 2 \langle 0.9848, -0.1736 \rangle = \langle 1.9696, -0.3472 \rangle \] 3. **Add the vectors component-wise:** \[ \mathbf{u} + \mathbf{v} = \langle 2.2944 + 1.9696, 3.2768 + (-0.3472) \rangle = \langle 4.264, 2.9296 \rangle \] 4. **Find the magnitude of the sum vector:** \[ \| \mathbf{u} + \mathbf{v} \| = \sqrt{4.264^2 + 2.9296^2} = \sqrt{18.179 + 8.584} = \sqrt{26.763} \approx 5.174 \] Rounded to the nearest tenth: \[ 5.2 \] 5. **Find the direction angle \( \theta \):** \[ \theta = \tan^{-1} \left( \frac{2.9296}{4.264} \right) = \tan^{-1}(0.6869) \approx 34.6^\circ \] Since both components are positive, \( \theta = 34.6^\circ \) is in the first quadrant, so no adjustment is needed. Rounded to the nearest degree: \[ 35^\circ \] 6. **Write the final answer in magnitude and direction form:** \[ \boxed{5.2 \langle \cos 35^\circ, \sin 35^\circ \rangle} \] This means the sum vector has magnitude approximately 5.2 and direction 35 degrees measured counterclockwise from the positive x-axis.