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The question seems to be focused on finding the parametric equations of a line, given a point and a direction vector, and determining where the line intersects a plane.

Let me extract and interpret the mathematical content based on your image.

1. **Problem:**
   - A line passes through the point \( P(1, 1, 2) \) and is parallel to the vector \( v = 2i + j - k \).
   - You are asked to write the parametric equations of the line.
   - Additionally, you are asked to find the point where the line intersects the given plane, described by the equation \( 3x + y - 4z = -2 \).

2. **Parametric Equations:**
   To find the parametric equations of a line, we use the formula:
   \[
   x = x_0 + at, \quad y = y_0 + bt, \quad z = z_0 + ct
   \]
   where \( (x_0, y_0, z_0) \) is the point on the line (here, \( P(1, 1, 2) \)) and \( (a, b, c) \) are the components of the direction vector \( v \). In this case, \( v = (2, 1, -1) \), so:

   \[
   x = 1 + 2t, \quad y = 1 + t, \quad z = 2 - t
   \]
   
3. **Intersection with the Plane:**
   The plane's equation is given as \( 3x + y - 4z = -2 \). To find the point of intersection, substitute the parametric equations into the plane equation:
   
   \[
   3(1 + 2t) + (1 + t) - 4(2 - t) = -2
   \]

   Simplify the equation step by step:
   \[
   3 + 6t + 1 + t - 8 + 4t = -2
   \]
   \[
   6t + t + 4t + 3 + 1 - 8 = -2
   \]
   \[
   11t - 4 = -2
   \]
   \[
   11t = 2
   \]
   \[
   t = \frac{2}{11}
   \]

4. **Substitute \( t = \frac{2}{11} \) back into the parametric equations:**
   - \( x = 1 + 2 \times \frac{2}{11} = 1 + \frac{4}{11} = \frac{15}{11} \)
   - \( y = 1 + \frac{2}{11} = \frac{13}{11} \)
   - \( z = 2 - \frac{2}{11} = \frac{20}{11} \)

Thus, the point where the line intersects the plane is \( \left( \frac{15}{11}, \frac{13}{11}, \frac{20}{11} \right) \).

Would you like more details on any of the steps, or do you have any questions?

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Here are 5 related questions to expand on this topic:

1. How do you derive the parametric equation of a line from two given points?
2. What is the geometric interpretation of a direction vector in 3D space?
3. How do you find the distance between a point and a plane in 3D?
4. How would the process change if the line was perpendicular to the plane instead of intersecting it?
5. How can you verify whether a point lies on a given line or plane?

**Tip:** When solving problems with parametric equations and planes, always check if your final solution satisfies both the line's parametric form and the plane equation to avoid calculation errors.

Write the parametric equation of the line L passing through point (1, 1, 2) and parallel to vector (2, 1, -1). Find where L intersects the plane given by the equation 3x + y - 4z = -2.

Learn how to write the parametric equation of a line passing through (1, 1, 2) and parallel to vector (2, 1, -1), and find the intersection with the plane 3x + y - 4z = -2. Suitable for pre-calculus and calculus students.

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