To find the vector and parametric equations of the line passing through the point $P(2, -3, 7)$ and parallel to the vector $\mathbf{v} = \langle 1, 3, -2 \rangle$, we follow these steps:

1. Vector Equation of the Line

The vector equation of a line is given by: $\mathbf{r}(t) = \mathbf{r}_0 + t \mathbf{v}$ Where:

• $\mathbf{r}_0$ is a position vector of a point on the line (here, $(2, -3, 7)$).
• $t$ is the parameter.
• $\mathbf{v}$ is the direction vector (here, $\langle 1, 3, -2 \rangle$).

Substitute the given values: $\mathbf{r}(t) = \langle 2, -3, 7 \rangle + t \langle 1, 3, -2 \rangle$ This gives the vector equation: $\mathbf{r}(t) = \langle 2 + t, -3 + 3t, 7 - 2t \rangle$

2. Parametric Equations of the Line

The parametric equations of the line are obtained by writing the components of the vector equation: $x(t) = 2 + t$ $y(t) = -3 + 3t$ $z(t) = 7 - 2t$

Thus, the parametric equations are: $x = 2 + t, \quad y = -3 + 3t, \quad z = 7 - 2t$

Summary:

• Vector equation: $\mathbf{r}(t) = \langle 2 + t, -3 + 3t, 7 - 2t \rangle$
• Parametric equations: $x = 2 + t, \quad y = -3 + 3t, \quad z = 7 - 2t$

Would you like further clarification on how we obtained these equations or any other details?

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Additional Questions:

1. How do you derive the direction vector from two points on a line?
2. What is the difference between the vector and parametric equations of a line?
3. Can the direction vector of a line be scaled? If so, how does that affect the line's equation?
4. How would the equation change if the line were not parallel to the given vector but orthogonal to it?
5. How do you find the intersection of two lines in 3D space?

Tip:

The vector equation represents the line in terms of a position vector plus a scalar multiple of the direction vector, while the parametric form breaks this into individual equations for $x$, $y$, and $z$.