Find both the vector equation and the parametric equations of the line through \( (0,0,0) \) that is parallel to the line \( \mathbf{r}=\langle 3-2 t, 9+5 t, 2+3 t\rangle \), where \( t=0 \) corresponds to the given point. The vector equation is \( \langle x, y, z\rangle=\langle\square, \square+\square \)

To find both the vector equation and the parametric equations of the line through the origin \( (0,0,0) \) that is parallel to the given line \[ \mathbf{r} = \langle 3 - 2t, 9 + 5t, 2 + 3t \rangle, \] follow these steps: ### 1. **Determine the Direction Vector** The given line can be written in the form: \[ \mathbf{r} = \mathbf{r}_0 + t \mathbf{v}, \] where: - \( \mathbf{r}_0 = \langle 3, 9, 2 \rangle \) is a point on the line, - \( \mathbf{v} = \langle -2, 5, 3 \rangle \) is the direction vector. Since the desired line is parallel to the given line, it will have the same direction vector \( \mathbf{v} = \langle -2, 5, 3 \rangle \). ### 2. **Vector Equation of the Desired Line** The vector equation of a line passing through the origin \( (0,0,0) \) with direction vector \( \mathbf{v} \) is: \[ \langle x, y, z \rangle = \langle 0, 0, 0 \rangle + t \langle -2, 5, 3 \rangle. \] Simplifying, we get: \[ \langle x, y, z \rangle = \langle -2t, 5t, 3t \rangle. \] ### 3. **Parametric Equations of the Desired Line** From the vector equation, we can extract the parametric equations by equating each component: \[ \begin{cases} x = -2t, \\ y = 5t, \\ z = 3t. \end{cases} \] Here, \( t \) is a real number parameter. ### **Summary** - **Vector Equation:** \[ \langle x, y, z \rangle = \langle 0, 0, 0 \rangle + t \langle -2, 5, 3 \rangle \quad \text{or} \quad \langle x, y, z \rangle = \langle -2t, 5t, 3t \rangle. \] - **Parametric Equations:** \[ \begin{cases} x = -2t, \\ y = 5t, \\ z = 3t. \end{cases} \] These equations define the line through the origin that is parallel to the given line. **Final Answer:** After simplifying, the vector equation is  ⟨x, y, z〉 = ⟨0, 0, 0〉 + t 〈–2, 5, 3〉 and the parametric equations are  x = –2 t,  y = 5 t,  z = 3 t.