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The document you uploaded contains exercises from a mathematics course, specifically dealing with topics like intersections of lines and planes, groups, vector spaces, and subspaces in \( \mathbb{R}^2 \). Here's a breakdown of the problems:

### Problem Overview:
1. **Intersection of a Plane and a Line**:  
   Find the intersection point between the plane \( E \) and the line \( G \) defined by parametric equations.

2. **Intersection of Lines in \( \mathbb{R}^2 \)**:  
   Determine intersection points of the lines \( g_1 \), \( g_2 \), \( g_3 \), and \( g_4 \) given by their equations or parametric forms.

3. **Group of Bijective Mappings**:  
   Prove that the set of bijective mappings \( S_n \), under composition and identity mapping, forms a group, and determine if it is Abelian.

4. **Permutations in \( S_3 \)**:  
   Given specific permutations \( \alpha \) and \( \beta \) in \( S_3 \), verify certain group properties and simplify expressions involving these permutations.

5. **Vector Space of Continuous Functions**:  
   Prove that the set of continuous functions with defined vector addition and scalar multiplication forms a vector space.

6. **Subspaces of \( \mathbb{R}^2 \)**:  
   Identify which of the provided sets are subspaces of \( \mathbb{R}^2 \) by checking the necessary axioms.

Would you like detailed solutions for any of these problems? Here are some related questions to consider:

1. How do you find the intersection of a line and a plane in 3D space?
2. What conditions are necessary for two lines in \( \mathbb{R}^2 \) to intersect?
3. What are the group properties that need to be checked to prove \( S_n \) forms a group?
4. How can the group operations of \( S_3 \) be visualized with permutations?
5. What axioms define a vector space, and how are they checked in different sets?

**Tip:** For working with group properties, visualizing elements and operations with diagrams (like permutations on a triangle) can greatly enhance understanding.

The document you uploaded contains exercises from a mathematics course, specifically dealing with topics like intersections of lines and planes, groups, vector spaces, and subspaces in \( \mathbb{R}^2 \).

Explore detailed solutions for exercises on intersections of lines and planes, group theory, and vector spaces from an undergraduate-level mathematics course. Topics covered include finding intersection points in 3D, verifying group properties, and identifying subspaces in \( \mathbb{R}^2 \).

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