Given the vector \( \mathbf{u} \) equal to \( 2\left\langle\cos 115^{\circ}, \sin 115^{\circ}\right\rangle \) and vector \( \mathbf{v} \) equal to \( 6\left\langle\cos 325^{\circ}, \sin 325^{\circ}\right\rangle \), find the sum \( \mathbf{u}+\mathbf{v} \) and write your answer in magnitude and direction form with the magnitude rounded to the nearest tenth and the direction rounded to the nearest degree, \( 0^{\circ} \leq \theta<360^{\circ} \).

To find the sum of the vectors \( \mathbf{u} \) and \( \mathbf{v} \), we first need to calculate each vector's components. 1. **Calculating \( \mathbf{u} \)**: \[ \mathbf{u} = 2\left\langle\cos 115^{\circ}, \sin 115^{\circ}\right\rangle \] The components are: \[ u_x = 2\cos 115^{\circ} \quad \text{and} \quad u_y = 2\sin 115^{\circ} \] Using a calculator: \[ u_x \approx 2 \times (-0.4226) \approx -0.8452 \] \[ u_y \approx 2 \times 0.9063 \approx 1.8126 \] Thus, \( \mathbf{u} \approx \left\langle -0.8452, 1.8126 \right\rangle \). 2. **Calculating \( \mathbf{v} \)**: \[ \mathbf{v} = 6\left\langle\cos 325^{\circ}, \sin 325^{\circ}\right\rangle \] The components are: \[ v_x = 6\cos 325^{\circ} \quad \text{and} \quad v_y = 6\sin 325^{\circ} \] Using a calculator: \[ v_x \approx 6 \times (0.9063) \approx 5.4378 \] \[ v_y \approx 6 \times (-0.4226) \approx -2.5356 \] Thus, \( \mathbf{v} \approx \left\langle 5.4378, -2.5356 \right\rangle \). 3. **Adding the vectors**: \[ \mathbf{u} + \mathbf{v} = \left\langle -0.8452 + 5.4378, 1.8126 - 2.5356 \right\rangle \] This gives: \[ x_{total} \approx 4.5926, \quad y_{total} \approx -0.7230 \] 4. **Magnitude**: \[ \text{Magnitude (M)} = \sqrt{x_{total}^2 + y_{total}^2} \approx \sqrt{(4.5926)^2 + (-0.7230)^2} \approx \sqrt{21.0791 + 0.5227} \approx \sqrt{21.6018} \approx 4.6469 \] Rounding to the nearest tenth, \( M \approx 4.6 \). 5. **Direction**: \[ \theta = \tan^{-1}\left(\frac{y_{total}}{x_{total}}\right) = \tan^{-1}\left(\frac{-0.7230}{4.5926}\right) \] This gives approximately \( -8.9^{\circ} \). Since the vector is in the fourth quadrant, we add \( 360^{\circ} \): \[ \theta \approx 360^{\circ} - 8.9^{\circ} \approx 351.1^{\circ} \] Rounding to the nearest degree, \( \theta \approx 351^{\circ} \). Thus, the sum \( \mathbf{u} + \mathbf{v} \) in magnitude and direction form is: \[ \text{Magnitude} \approx 4.6, \quad \text{Direction} \approx 351^{\circ}. \]