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.