Compute the dot product of the vectors \( \mathbf{u} \) and \( \mathbf{v} \), and find the angle between the vectors. \( \mathbf{u}=\langle-12,0,5\rangle \) 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.)

Let's compute the dot product, magnitudes of the vectors, and the angle between them step by step. ### Given Vectors: \[ \mathbf{u} = \langle -12, 0, 5 \rangle \quad \text{and} \quad \mathbf{v} = \langle 1, 2, 3 \rangle \] ### 1. Dot Product (\( \mathbf{u} \cdot \mathbf{v} \)) The dot product is calculated as: \[ \mathbf{u} \cdot \mathbf{v} = (-12)(1) + (0)(2) + (5)(3) = -12 + 0 + 15 = 3 \] \[ \mathbf{u} \cdot \mathbf{v} = \boxed{3} \] ### 2. Magnitudes of the Vectors - **Magnitude of \( \mathbf{u} \)**: \[ |\mathbf{u}| = \sqrt{(-12)^2 + 0^2 + 5^2} = \sqrt{144 + 0 + 25} = \sqrt{169} = 13 \] \[ |\mathbf{u}| = \boxed{13} \] - **Magnitude of \( \mathbf{v} \)**: \[ |\mathbf{v}| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14} \] \[ |\mathbf{v}| = \boxed{\sqrt{14}} \] ### 3. Angle Between the Vectors (\( \theta \)) The angle \( \theta \) between the vectors can be found using the dot product formula: \[ \mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| \cdot |\mathbf{v}| \cdot \cos{\theta} \] Solving for \( \cos{\theta} \): \[ \cos{\theta} = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| \cdot |\mathbf{v}|} = \frac{3}{13 \cdot \sqrt{14}} \approx \frac{3}{48.6415} \approx 0.0617 \] Now, find \( \theta \) by taking the inverse cosine: \[ \theta = \arccos(0.0617) \approx 86.46^\circ \] \[ \theta = \boxed{86.46^\circ} \] ### Summary - **Dot Product**: \( \mathbf{u} \cdot \mathbf{v} = 3 \) - **Magnitudes**: - \( |\mathbf{u}| = 13 \) - \( |\mathbf{v}| = \sqrt{14} \) - **Angle Between Vectors**: \( 86.46^\circ \)