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The given task consists of describing the vertices and edges of both the cube and the octahedron inscribed within the cube, using vectors. Let's break this down step by step:

### a) **Cube:**
1. **Vertices:**
   The cube has edge length \( 2E \), with the vertex \( O \) positioned at the origin \((0,0,0)\) in the coordinate system. We can express the coordinates of the vertices of the cube in terms of \( E \), the half-length of the cube.

   - \( O = (0, 0, 0) \)
   - \( A = (0, 0, 2E) \)
   - \( B = (0, 2E, 2E) \)
   - \( C = (2E, 2E, 2E) \)
   - \( D = (2E, 0, 2E) \)
   - \( E = (0, 0, E) \) (Side midpoint of cube face)
   - \( F = (2E, 2E, 0) \) etc.

   There will be eight main cube vertices and other points such as the midpoints which define the octahedron.

2. **Edges:**
   The edges of the cube connect these vertices. The edges can be expressed as vectors in terms of \( E \). For example:
   - Vector from \( O \) to \( A \): \( \vec{OA} = (0, 0, 2E) \)
   - Vector from \( A \) to \( B \): \( \vec{AB} = (0, 2E, 0) \)
   - Vector from \( B \) to \( C \): \( \vec{BC} = (2E, 0, 0) \)
   - Vector from \( O \) to \( D \): \( \vec{OD} = (2E, 0, 0) \)
   
   The cube has 12 edges in total, each represented by similar vector combinations.

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### b) **Octahedron:**
1. **Vertices:**
   The vertices of the octahedron lie at the midpoints of the faces of the cube. These vertices can also be described by vectors, which are essentially the averages of the cube's face vertices. The vertices are as follows:
   - Vertex on \( x \)-face midpoint: \( (E, 0, E) \)
   - Vertex on \( y \)-face midpoint: \( (0, E, E) \)
   - Vertex on \( z \)-face midpoint: \( (E, E, 0) \)
   - Etc.

   The octahedron will have 6 vertices in total, one for each face of the cube.

2. **Edges:**
   The edges of the octahedron connect these midpoints. We can express these edges as vectors. For instance:
   - Edge from \( (E, 0, E) \) to \( (0, E, E) \): \( \vec{E_1E_2} = (-E, E, 0) \)
   - Edge from \( (0, E, E) \) to \( (E, E, 0) \): \( \vec{E_2E_3} = (E, 0, -E) \)

   The octahedron will have 12 edges, each corresponding to a vector connection between these face midpoints.

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Would you like more detailed explanations or have any specific questions about this solution?

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Here are 5 related questions that could further extend the topic:
1. How do the diagonal vectors of the cube relate to the vectors of the octahedron?
2. What is the formula for the length of the edges of the octahedron in terms of \( E \)?
3. How do the symmetries of the cube and the octahedron compare geometrically?
4. What is the geometric relationship between the cube's diagonals and the octahedron's vertices?
5. How can we derive the volume of the octahedron given the edge length of the cube?

**Tip:** Visualizing the cube in a coordinate system helps in better understanding vector descriptions and symmetries!

Describe the vertices and edges of the cube and the inscribed octahedron using vectors.

Learn how to describe the vertices and edges of a cube with edge length 2E and an inscribed octahedron using vectors. This problem covers key concepts in vector geometry and the symmetry of polyhedra. Suitable for students in Grades 9-12.

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