The image shows a question involving planes, lines, and vectors in three-dimensional geometry. Let's break it down and provide the solution.

Problem Breakdown:

1. Given:

   • The problem involves two planes, $P_1$ and $P_2$.
   • The planes are given as $P_1: 2x + y - z = 1$ and $P_2: 3x - 2y + z = 4$.
   • Points $A(0, 0, 0)$ and $B(3, 2, 4)$ are on the planes.

2. Task:

   • The question asks for the position of a line in the planes.

3. Options:

   • (1) Line $l$ lies in both planes $P_1$ and $P_2$.
   • (2) Line $l$ is parallel to planes $P_1$ and $P_2$.
   • (3) Line $l$ is perpendicular to both planes $P_1$ and $P_2$.
   • (4) Planes $P_1$ and $P_2$ are coplanar.

Solution:

To solve this problem, consider the following:

• Intersection of Two Planes:

  • The intersection of two planes $P_1$ and $P_2$ is a line if the planes are not parallel.
  • This line $l$ will lie in both planes.

• Check if Planes are Parallel:

  • Compare the normal vectors of $P_1$ and $P_2$:
    • For $P_1: 2x + y - z = 1$, the normal vector $\vec{n_1}$ is $(2, 1, -1)$.
    • For $P_2: 3x - 2y + z = 4$, the normal vector $\vec{n_2}$ is $(3, -2, 1)$.
  • These vectors are not scalar multiples of each other, meaning the planes are not parallel.

• Conclusion:

  • Since the planes intersect and are not parallel, the line $l$ will lie in both planes.

Correct Answer:

• The correct choice is (1): The line $l$ lies in both planes $P_1$ and $P_2$.

Would you like a more detailed explanation or have any questions?

Related Questions:

1. How can you find the equation of the line of intersection between two planes?
2. What conditions must two planes satisfy to be parallel?
3. How do you determine if two planes are perpendicular?
4. How can the distance between two parallel planes be calculated?
5. What is the geometric interpretation of the dot product of two vectors?

Tip:

Always remember that the intersection of two non-parallel planes is a line, which can be found by solving the system of equations given by the planes.