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We are given the following problem:

- We need to find a vector equation and parametric equations for a line.
- The line passes through the point \( (0, 15, -10) \).
- The line is parallel to the line represented by the parametric equations:
  \[
  x = -1 + 3t, \quad y = 6 - 2t, \quad z = 3 + 7t.
  \]

### Steps to solve:

1. **Direction Vector of the Line:**
   The line is parallel to another line, meaning both lines share the same direction vector. The parametric equations for the given line are:
   \[
   x = -1 + 3t, \quad y = 6 - 2t, \quad z = 3 + 7t.
   \]
   From this, the direction vector can be extracted as \( \mathbf{d} = \langle 3, -2, 7 \rangle \), which represents the coefficients of \( t \) in each parametric equation.

2. **Vector Equation of the Line:**
   The vector equation of a line can be written as:
   \[
   \mathbf{r}(t) = \mathbf{r}_0 + t \mathbf{d}
   \]
   where \( \mathbf{r}_0 \) is the position vector of a point on the line, and \( \mathbf{d} \) is the direction vector.
   
   Here, \( \mathbf{r}_0 = \langle 0, 15, -10 \rangle \) (the given point), and \( \mathbf{d} = \langle 3, -2, 7 \rangle \).

   Therefore, the vector equation is:
   \[
   \mathbf{r}(t) = \langle 0, 15, -10 \rangle + t \langle 3, -2, 7 \rangle
   \]
   Simplified, this becomes:
   \[
   \mathbf{r}(t) = \langle 3t, 15 - 2t, -10 + 7t \rangle.
   \]

3. **Parametric Equations:**
   From the vector equation, we can write the parametric equations for the line:
   \[
   x(t) = 3t, \quad y(t) = 15 - 2t, \quad z(t) = -10 + 7t.
   \]

### Final Answer:

- **Vector equation:**
  \[
  \mathbf{r}(t) = \langle 0, 15, -10 \rangle + t \langle 3, -2, 7 \rangle
  \]
  or equivalently,
  \[
  \mathbf{r}(t) = \langle 3t, 15 - 2t, -10 + 7t \rangle.
  \]

- **Parametric equations:**
  \[
  x(t) = 3t, \quad y(t) = 15 - 2t, \quad z(t) = -10 + 7t.
  \]

Would you like any additional details or further clarifications?

### Related Questions:
1. How do you find the direction vector from a parametric equation?
2. What is the general form of a vector equation for a line?
3. How do you determine if two lines are parallel based on their parametric equations?
4. What are the parametric equations of a line in 3D space?
5. How can you convert a vector equation into parametric form?

### Tip:
Always ensure the direction vector of the line is correctly extracted from the parametric equations, as it directly affects both the vector and parametric forms of the solution.

Find a vector equation and parametric equations for the line passing through the point (0, 15, -10) and parallel to the line with parametric equations: x = -1 + 3t, y = 6 - 2t, z = 3 + 7t.

Learn how to find the vector and parametric equations for a line passing through the point (0, 15, -10) and parallel to the line with parametric equations x = -1 + 3t, y = 6 - 2t, z = 3 + 7t. This problem demonstrates key concepts in 3D geometry and vector math.

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