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.