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.)

To find the angle between the vectors \( \mathbf{u} \) and \( \mathbf{v} \), we use the formula for the dot product: \[ \mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos \theta \] Given: - \( \mathbf{u} \cdot \mathbf{v} = 3 \) - \( |\mathbf{u}| = 13 \) - \( |\mathbf{v}| = \sqrt{14} \) We can rearrange this formula to solve for \( \cos \theta \): \[ \cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| |\mathbf{v}|} \] Plugging in the values: \[ \cos \theta = \frac{3}{13 \cdot \sqrt{14}} \] Calculating \( 13 \cdot \sqrt{14} \): \[ 13 \cdot \sqrt{14} = 13\sqrt{14} \] So we have: \[ \cos \theta = \frac{3}{13\sqrt{14}} \] To find \( \theta \), we take the inverse cosine: \[ \theta = \cos^{-1}\left(\frac{3}{13\sqrt{14}}\right) \] Now, let's calculate this using a calculator: 1. Calculate \( 13\sqrt{14} \) which is approximately \( 48.541 \). 2. Now, calculate \( \frac{3}{48.541} \) which is approximately \( 0.0618 \). 3. Finally, use the inverse cosine function: \[ \theta \approx \cos^{-1}(0.0618) \approx 85.44^{\circ} \] So the angle between the vectors is approximately: \[ \theta \approx 85.44^{\circ} \] Thus, rounding to the nearest hundredth: \[ \theta \approx 85.44^{\circ} \]