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### **Aufgabe 1** (Problem 1):
You are asked to provide a normal equation and a possible coordinate equation for different planes.

1. **Part a**:  
   The plane \( e_1 \) is parallel to the \( x_1 x_3 \)-plane and passes through the point \( P(0|4|0) \).  
   - A plane parallel to the \( x_1 x_3 \)-plane has a normal vector parallel to the \( x_2 \)-axis, i.e., the normal vector is \( \mathbf{n} = (0, 1, 0) \).  
   - Using the point-normal form of a plane equation: \( n_1(x - x_0) + n_2(y - y_0) + n_3(z - z_0) = 0 \), we get the equation for this plane as:  
     \[
     1 \cdot (x_2 - 4) = 0 \quad \text{or simply} \quad x_2 = 4.
     \]

2. **Part b**:  
   The plane \( e_2 \) is perpendicular to the \( x_1 \)-axis and contains the point \( P(1|2|3) \).  
   - A plane perpendicular to the \( x_1 \)-axis has a normal vector \( \mathbf{n} = (1, 0, 0) \).  
   - Using the same form of equation as above and the given point, the equation for this plane is:  
     \[
     1 \cdot (x_1 - 1) = 0 \quad \text{or simply} \quad x_1 = 1.
     \]

3. **Part c**:  
   The plane \( e_3 \) passes through the origin and is perpendicular to the line passing through the points \( P(3|2|1) \) and \( Q(2|0|1) \).  
   - The direction vector of the line through \( P \) and \( Q \) is \( \vec{d} = P - Q = (1, 2, 0) \).  
   - The normal vector of the plane is perpendicular to this direction vector, so we can use it in the cross-product to form the equation of the plane. Using the given points and the general plane equation method will give you the result.

### **Aufgabe 2** (Problem 2):
Here, you're asked to define a plane by a point and a line, not just by three points.

1. **Part a**:  
   To define a plane uniquely by a point \( P \) and a line \( g: \vec{x} = \vec{a} + \lambda \cdot \vec{u} \), the point \( P \) must **not** lie on the line \( g \). Also, the vector connecting the point \( P \) and any point on the line must not be parallel to the direction vector of the line \( \vec{u} \). The condition for this is that the vector \( \vec{P} - \vec{a} \) must not be collinear with \( \vec{u} \).

2. **Part b**:  
   The equation of the plane in parametric form can be written as:  
   \[
   \vec{x} = \vec{a} + \lambda \vec{u} + \mu (\vec{P} - \vec{a}).
   \]

### **Aufgabe 3** (Problem 3):
The task is to match the given diagrams of planes to their corresponding equations.

- **Diagram 1**: The plane is vertical along \( x_2 \) and passes through a certain area in the \( x_1 x_3 \)-plane. A likely equation is:  
  \( 4x_1 - 3x_3 = 12 \) (option a).

- **Diagram 2**: The plane is vertical along the \( x_3 \)-axis and has an intercept on the \( x_2 \)-axis. A likely equation is:  
  \( x_2 - 4x_3 = 8 \) (option b).

- **Diagram 3**: The plane is parallel to the \( x_2 x_3 \)-plane and cuts the \( x_1 \)-axis at a specific point. A possible equation is:  
  \( x_3 = 2 \) (option c).

- **Diagram 4**: The plane is parallel to the \( x_1 x_2 \)-plane and lies at a fixed height on the \( x_3 \)-axis. A possible equation is:  
  \( x_3 = 0 \) (option f).

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Let me know if you need detailed clarifications on any of these points!

Here are 5 related questions for further practice:
1. How do you derive the normal vector of a plane from a given line and point?
2. Can a plane be uniquely determined by two parallel lines?
3. How do you find the intersection line of two planes?
4. How can you generalize the equation of a plane passing through three points?
5. What are the conditions for two planes to be perpendicular to each other?

**Tip:** Always remember to check if vectors are parallel or perpendicular when working with planes and lines.

Solve the given tasks related to plane equations and lines in 3D space.

Learn how to solve tasks involving 3D plane equations, normal vectors, and line-plane intersections. This problem explores core vector calculus concepts and parametric plane equations, suitable for university-level mathematics students.

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