The image you uploaded shows a cube-related exercise in German, involving vectors and coordinates. Here's a translation and breakdown of the problem for further analysis:

Problem 8: A cube whose edges are parallel to the coordinate axes has an edge length of 5, and the origin of the coordinate system is the center of the base.

1. (a) Specify the coordinates of the vertices of the cube.

2. (b) Determine the equations for the lines on which the space diagonals $AG$ and $BH$ of the cube lie.

There are some vector equations provided:

• For $AG$, the parametric line equation is: $\vec{x} = \vec{A} + r \cdot \overrightarrow{AG}$

• For $BH$, the parametric line equation is: $\vec{x} = \vec{B} + r \cdot \overrightarrow{BH}$

Let’s proceed step by step to solve this:

Step 1: Determine the Coordinates of the Cube's Vertices (Part a)

The cube is centered at the origin, and its edges are parallel to the coordinate axes with edge length 5. We need to find the coordinates for all vertices, typically labeled $A, B, C, D, E, F, G, H$.

If the center of the base is at the origin, the vertices can be calculated as follows:

1. The base is in the $xy$-plane, so its vertices have $z = 0$.
2. The top of the cube is parallel to the base but displaced by the edge length along the $z$-axis.

Given that the edge length is 5, the vertices would be symmetric around the origin. Let’s assume the following coordinates:

• $A(-2.5, -2.5, 0)$
• $B(2.5, -2.5, 0)$
• $C(2.5, 2.5, 0)$
• $D(-2.5, 2.5, 0)$
• $E(-2.5, -2.5, 5)$
• $F(2.5, -2.5, 5)$
• $G(2.5, 2.5, 5)$
• $H(-2.5, 2.5, 5)$

Step 2: Determine the Equations of the Space Diagonals (Part b)

For the space diagonals $AG$ and $BH$:

• $AG$ connects the points $A(-2.5, -2.5, 0)$ and $G(2.5, 2.5, 5)$. The vector form of this line is given by: $\overrightarrow{AG} = \vec{G} - \vec{A} = (2.5 - (-2.5), 2.5 - (-2.5), 5 - 0) = (5, 5, 5)$ Therefore, the parametric equation is: $\vec{x} = (-2.5, -2.5, 0) + r(5, 5, 5)$ where $r \in \mathbb{R}$.

• $BH$ connects the points $B(2.5, -2.5, 0)$ and $H(-2.5, 2.5, 5)$. The vector form of this line is: $\overrightarrow{BH} = \vec{H} - \vec{B} = (-2.5 - 2.5, 2.5 - (-2.5), 5 - 0) = (-5, 5, 5)$ Therefore, the parametric equation is: $\vec{x} = (2.5, -2.5, 0) + r(-5, 5, 5)$ where $r \in \mathbb{R}$.

Would you like me to expand further on these steps or clarify anything?

Here are 5 related questions to explore:

1. How are parametric equations for lines in 3D generally constructed?
2. How would you calculate the length of the space diagonal of the cube?
3. What is the equation for a line passing through two points in 3D?
4. How do space diagonals relate to the geometry of a cube?
5. Can you generalize this process for cubes of different edge lengths?

Tip: Remember that space diagonals in a cube always have equal lengths, regardless of orientation.