Write the vector \( \mathbf{v}=\langle-6,-3,-6\rangle \) as a product of its magnitude and a unit vector with the same direction as \( \mathbf{v} \). \( \langle-6,-3,-6\rangle=\square\langle(\square,(\square),(\square)\rangle \) (Simplify your answers. Use integers or fractions for any numbers in the expression.)

To express the vector \( \mathbf{v} = \langle -6, -3, -6 \rangle \) as a product of its magnitude and a unit vector in the same direction, follow these steps: 1. **Calculate the Magnitude of \( \mathbf{v} \):** \[ \|\mathbf{v}\| = \sqrt{(-6)^2 + (-3)^2 + (-6)^2} = \sqrt{36 + 9 + 36} = \sqrt{81} = 9 \] 2. **Determine the Unit Vector in the Direction of \( \mathbf{v} \):** Divide each component of \( \mathbf{v} \) by its magnitude: \[ \mathbf{u} = \left\langle \frac{-6}{9}, \frac{-3}{9}, \frac{-6}{9} \right\rangle = \left\langle -\frac{2}{3}, -\frac{1}{3}, -\frac{2}{3} \right\rangle \] 3. **Express \( \mathbf{v} \) as the Product of Its Magnitude and the Unit Vector:** \[ \mathbf{v} = 9 \left\langle -\frac{2}{3}, -\frac{1}{3}, -\frac{2}{3} \right\rangle \] **Final Answer:** \[ \langle -6, -3, -6 \rangle = 9\,\langle\, -\tfrac{2}{3},\; -\tfrac{1}{3},\; -\tfrac{2}{3} \,\rangle \]