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Let's tackle the exercises step-by-step.

### Exercise 5
1. **Placer les points M et J**: This requires plotting the points \( M \) and \( J \) based on their coordinates.
2. **Décomposer le vecteur \(\overrightarrow{JM}\)** en fonction des vecteurs \(\overrightarrow{E}\) et \(\overrightarrow{G}\): This involves expressing \(\overrightarrow{JM}\) as a linear combination of the given vectors.
3. **En déduire la position relative de la droite (HJ) et du plan (EGF)**: After finding the vector form, analyze the alignment of line (HJ) concerning plane (EGF) to determine if they intersect, are parallel, etc.

### Exercise 6
1. **Les points A, B et C définissent-ils un plan?**: Check if points \( A(1, -1, 2), B(3, 3, 8), C(-3, 5, 4) \) form a plane by calculating if they are non-collinear.
2. **Les points A, C et D sont alignés**: Determine if points \( A, C, \) and \( D(1, 2, 3) \) are collinear by checking if vectors \(\overrightarrow{AC}\) and \(\overrightarrow{AD}\) are parallel.
3. **Le point D est le milieu du segment [AB]**: Verify if \( D \) is the midpoint by comparing it to the midpoint formula between \( A \) and \( B \).
4. **La droite (DE) et le plan (ABC) ont un unique point commun**: Analyze if line (DE), formed by points \( D \) and \( E(4, -1, 9) \), intersects the plane formed by points \( A, B, \) and \( C \).

Would you like to delve into specific calculations for any of these steps?

### Questions for Further Exploration
1. How do you determine if points are collinear in three-dimensional space?
2. What is the method to find a point’s coordinates on a line segment?
3. How do you check if a point lies on a plane?
4. What criteria define the intersection between a line and a plane?
5. How can vector decomposition help in spatial analysis?

**Tip**: For collinearity, ensure that one vector is a scalar multiple of another when examining alignment.

Exercises involving points, vectors, planes, and line intersections in a 3D coordinate system.

Explore 3D geometry exercises involving points, planes, vector decomposition, and intersection analysis. These exercises cover essential concepts like collinearity, coplanarity, and conditions for points and lines in a 3D coordinate system. Suitable for Grade 12 and university students.

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