Given the vector \( \mathbf{u} \) equal to \( 3\left\langle\cos 325^{\circ}, \sin 325^{\circ}\right\rangle \) and vector \( \mathbf{v} \) equal to \( 5\left\langle\cos 40^{\circ}, \sin 40^{\circ}\right\rangle \), 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 \( \mathbf{u} + \mathbf{v} \), we first need to express both vectors in Cartesian (rectangular) coordinates. For vector \( \mathbf{u} \): \[ \mathbf{u} = 3\left\langle\cos 325^{\circ}, \sin 325^{\circ}\right\rangle = \left\langle 3\cos 325^{\circ}, 3\sin 325^{\circ} \right\rangle \] Calculating \( \cos 325^{\circ} \) and \( \sin 325^{\circ} \): \[ \cos 325^{\circ} \approx 0.8192, \quad \sin 325^{\circ} \approx -0.5736 \] Thus: \[ \mathbf{u} \approx \left\langle 3 \times 0.8192, 3 \times -0.5736 \right\rangle \approx \left\langle 2.4576, -1.7208 \right\rangle \] For vector \( \mathbf{v} \): \[ \mathbf{v} = 5\left\langle\cos 40^{\circ}, \sin 40^{\circ}\right\rangle = \left\langle 5\cos 40^{\circ}, 5\sin 40^{\circ} \right\rangle \] Calculating \( \cos 40^{\circ} \) and \( \sin 40^{\circ} \): \[ \cos 40^{\circ} \approx 0.7660, \quad \sin 40^{\circ} \approx 0.6428 \] Thus: \[ \mathbf{v} \approx \left\langle 5 \times 0.7660, 5 \times 0.6428 \right\rangle \approx \left\langle 3.8300, 3.2140 \right\rangle \] Now, summing the vectors \( \mathbf{u} \) and \( \mathbf{v} \): \[ \mathbf{u} + \mathbf{v} \approx \left\langle 2.4576 + 3.8300, -1.7208 + 3.2140 \right\rangle \approx \left\langle 6.2876, 1.4932 \right\rangle \] Next, we compute the magnitude of \( \mathbf{u} + \mathbf{v} \): \[ |\mathbf{u} + \mathbf{v}| = \sqrt{(6.2876)^2 + (1.4932)^2} \approx \sqrt{39.4787 + 2.2287} \approx \sqrt{41.7074} \approx 6.4360 \text{ (rounded to 6.4)} \] Now for the direction (angle \( \theta \)): \[ \theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{1.4932}{6.2876}\right) \] Calculating: \[ \theta \approx \tan^{-1}(0.2377) \approx 13.3^{\circ} \text{ (in the first quadrant)} \] \(\theta\) can be rounded to the nearest degree as \( 13^{\circ} \). Thus, the sum \( \mathbf{u} + \mathbf{v} \) in magnitude and direction form is approximately: \[ \boxed{6.4 \text{ at } 13^{\circ}} \]