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 \( \mathbf{v}=\langle 1,2,3\rangle \). \( \mathbf{u} \cdot \mathbf{v}=\square \). (Type an integer or a simplified fraction.)

To compute the **dot product** of vectors \( \mathbf{u} \) and \( \mathbf{v} \), and find the **angle** between them, follow these steps: ### 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: \[ \mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + u_3 v_3 \] Plugging in the values: \[ \mathbf{u} \cdot \mathbf{v} = (-12)(1) + (0)(2) + (5)(3) = -12 + 0 + 15 = 3 \] \[ \boxed{ \mathbf{u} \cdot \mathbf{v} = 3 } \] --- ### 2. Angle Between Vectors (\( \theta \)): The angle \( \theta \) between two vectors is given by: \[ \cos \theta = \frac{ \mathbf{u} \cdot \mathbf{v} }{ \| \mathbf{u} \| \, \| \mathbf{v} \| } \] #### a. Magnitudes of \( \mathbf{u} \) and \( \mathbf{v} \): \[ \| \mathbf{u} \| = \sqrt{(-12)^2 + 0^2 + 5^2} = \sqrt{144 + 0 + 25} = \sqrt{169} = 13 \] \[ \| \mathbf{v} \| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14} \] #### b. Calculating \( \cos \theta \): \[ \cos \theta = \frac{3}{13 \times \sqrt{14}} = \frac{3}{13\sqrt{14}} \approx \frac{3}{48.5415} \approx 0.0618 \] #### c. Determining \( \theta \): \[ \theta = \cos^{-1}(0.0618) \approx 86.3^\circ \] \[ \boxed{ \theta \approx 86.3^\circ } \] --- ### **Summary:** - **Dot Product:** \( \mathbf{u} \cdot \mathbf{v} = 3 \) - **Angle Between Vectors:** \( \theta \approx 86.3^\circ \)