Part 2 of 3 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. \( |\mathbf{u}|=\square \) and \( |\mathbf{v}|=\square \) (Type exact answers, using radicals as needed.)

To compute the dot product of the vectors \( \mathbf{u} \) and \( \mathbf{v} \): 1. **Calculation of the Dot Product**: \[ \mathbf{u} \cdot \mathbf{v} = (-12) \cdot 1 + 0 \cdot 2 + 5 \cdot 3 = -12 + 0 + 15 = 3. \] Now, to find the magnitudes of the vectors \( |\mathbf{u}| \) and \( |\mathbf{v}| \): 2. **Magnitude of Vector \( \mathbf{u} \)**: \[ |\mathbf{u}| = \sqrt{(-12)^2 + 0^2 + 5^2} = \sqrt{144 + 0 + 25} = \sqrt{169} = 13. \] 3. **Magnitude of Vector \( \mathbf{v} \)**: \[ |\mathbf{v}| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14}. \] So, we have: \[ |\mathbf{u}| = 13 \quad \text{and} \quad |\mathbf{v}| = \sqrt{14}. \] Now, to find the angle \( \theta \) between the vectors, we use the formula: \[ \cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| |\mathbf{v}|}. \] Substituting the known values: \[ \cos(\theta) = \frac{3}{13 \cdot \sqrt{14}}. \] To find \( \theta \), take the arccosine: \[ \theta = \arccos\left(\frac{3}{13 \sqrt{14}}\right). \] At this point, you can use a calculator or tables to find the angle in degrees or radians as needed!