https://r2.erweima.ai/mathbot/image/17255105741768awl4ozf.pdf

The document uploaded contains a variety of analytical geometry exercises in Spanish, including vector proofs, coordinate transformations, and geometric constructions. Below are a few examples of the exercises:

1. **Prove**: ⟨x−y, x+y⟩ = ⟨x, x⟩ − ⟨y, y⟩
2. **Prove**: |x−y| + |x+y| ≤ 2(|x| + |y|)
3. In two coordinate systems, find the coordinates of a point A in one system based on its coordinates in another, and vice versa for point B.
4. Given transformations between two coordinate systems, find the coordinates of the origin \(O'\) and the vectors \(e'_1, e'_2, e'_3\) in the first system.

Let me know if you need help solving any of these problems or want further clarifications!

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Would you like a detailed solution to any specific problem? Here are some related questions:

1. How do you perform a vector transformation between coordinate systems?
2. What methods are used to prove inequalities involving vector norms?
3. What are the properties of vector dot products and their applications?
4. How do you find the midpoint of a geometric segment in 3D space?
5. What are the steps to solve transformations between affine coordinate systems?

**Tip:** When transforming coordinates, always carefully track the origin shifts and the basis vector transformations.

The document contains exercises from Analytical Geometry including vector proofs, coordinate transformations, and geometric constructions. Examples include proving inequalities, performing coordinate transformations, and finding the coordinates of geometric figures in different systems.

Explore advanced exercises in Analytical Geometry, including vector proofs and coordinate transformations. Learn to prove inequalities, find coordinates in different systems, and apply geometric principles in space. Ideal for undergraduate students studying vector algebra and coordinate systems.

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