Compute the dot product of the vectors \( u \) and \( v \), and find the angle between the vectors. \( u=\langle-12,0,5\rangle \) and \( \mathbf{v}=\langle 1,2,3\rangle \). \( \mathbf{u} \cdot \mathbf{v}=3 \). (Type an integer or a simplified fraction.) Find the magnitude of the vectors. \( |\mathbf{u}|=13 \) and \( |\mathbf{v}|=\sqrt{14} \) (Type exact answers, using radicals as needed.) The angle between the vectors is \( \square^{\circ} \). (Type your answer in degrees. Do not round until the final answer. Then round to the nearest hundredth as needed.)

To find the angle between the vectors \( \mathbf{u} = \langle -12, 0, 5 \rangle \) and \( \mathbf{v} = \langle 1, 2, 3 \rangle \), we follow these steps: 1. **Dot Product**: \[ \mathbf{u} \cdot \mathbf{v} = (-12)(1) + (0)(2) + (5)(3) = -12 + 0 + 15 = 3 \] 2. **Magnitudes**: \[ |\mathbf{u}| = \sqrt{(-12)^2 + 0^2 + 5^2} = \sqrt{144 + 0 + 25} = \sqrt{169} = 13 \] \[ |\mathbf{v}| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14} \] 3. **Finding the Angle**: The angle \( \theta \) between the vectors can be found using the formula: \[ \cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| |\mathbf{v}|} = \frac{3}{13 \sqrt{14}} \] Calculating the cosine value: \[ \cos \theta \approx \frac{3}{13 \times 3.7416573867739413} \approx \frac{3}{48.641545} \approx 0.0617 \] To find \( \theta \), take the arccosine of 0.0617: \[ \theta \approx \arccos(0.0617) \approx 86.44^\circ \] (This is calculated using a calculator with sufficient precision to ensure accuracy to the nearest hundredth.) **Final Answer**: The angle between the vectors is \( \boxed{86.44^\circ} \).