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The image you've uploaded contains a set of math problems involving vectors, parametric equations, and geometry, specifically focusing on lines and planes in three-dimensional space.

### Let's break down the problems shown:

1. **Aufgabe 1** (Problem 1): 
   - **Topic:** Writing parametric equations for a line.
   - **Instruction:** You are asked to provide the parametric equation of a line through two given points.
   - Example Points: 
     - a) \( A(0|0|1), B(2|1|2) \)
     - b) \( A(1|0|1), B(2|1|8) \)

2. **Aufgabe 2** (Problem 2): 
   - **Topic:** Investigating the relative positions of two lines.
   - **Instruction:** You are to determine whether the lines are intersecting, parallel, or skew. If they intersect, you're asked to provide the coordinates of the intersection point.
   - Several parametric equations for different lines are given, and you need to compare them.

3. **Aufgabe 3** (Problem 3): 
   - **Topic:** Determining the parametric equation of a plane given three points.
   - Example Points: 
     - a) \( A(3|2|-1), B(5|3|1), C(0|1|2) \)

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### Solution approach for **Problem 1a**:
You are asked to find the parametric equation of a line passing through points \( A(0,0,1) \) and \( B(2,1,2) \).

1. **Direction vector (AB)** is obtained by subtracting coordinates of \( A \) from \( B \):
   \[
   \overrightarrow{AB} = (B - A) = (2 - 0, 1 - 0, 2 - 1) = (2, 1, 1)
   \]

2. **Parametric equation** for the line:
   \[
   \vec{r}(t) = A + t \cdot \overrightarrow{AB}


The image contains three problems related to parametric equations, vectors, and geometry in 3D space. The problems focus on: 1. Writing parametric equations of a line through two given points. 2. Investigating the relative positions of two lines (parallel, intersecting, or skew). 3. Determining the parametric equation of a plane given three points.

Learn how to solve problems involving parametric equations of lines and planes in 3D space. This includes finding the parametric equation of a line through two points, investigating relative line positions, and determining the parametric equation of a plane given three points. Suitable for students in Grades 10-12.

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