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The exercises in the image cover the topic of vectors, lines, and planes in space. Let's break them down step by step to solve them.

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**Exercice 1:**

Given a tetrahedron \( ABCD \), with point \( I \) as the midpoint of segment \( [AD] \). Points \( G \) and \( H \) are defined as:

\[
\overrightarrow{BG} = \overrightarrow{AB} + \overrightarrow{AC}
\quad \text{and} \quad \overrightarrow{IH} = \frac{3}{2} \overrightarrow{AD}
\]

**1) Express \( \overrightarrow{GC} \) as a linear combination of \( \overrightarrow{AB}, \overrightarrow{AC}, \overrightarrow{AD} \).**

We have \( \overrightarrow{GC} = \overrightarrow{BC} - \overrightarrow{BG} \), and we know \( \overrightarrow{BG} \). Use vector addition and substitution to express \( \overrightarrow{GC} \).

**2) Express \( \overrightarrow{HC} \) as a linear combination of \( \overrightarrow{AB}, \overrightarrow{AC}, \overrightarrow{AD} \).**

Here, \( \overrightarrow{HC} = \overrightarrow{IC} - \overrightarrow{IH} \), and using the midpoint property \( I \), express the vector as a combination.

**3) Decompose \( \overrightarrow{GH} \) in the basis \( (\overrightarrow{AB}, \overrightarrow{AC}, \overrightarrow{AD}) \).**

Use the definitions of \( G \) and \( H \) to find the vector \( \overrightarrow{GH} \) and decompose it accordingly.

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**Exercice 2:**

You are given:
- Point \( A(-1, 2, 3) \)
- Vectors \( \overrightarrow{u}(1, 0, 3) \) and \( \overrightarrow{v}(0, 1, 3) \)
- Plane \( P \) defined by \( A \) and the vectors \( \overrightarrow{u} \) and \( \overrightarrow{v} \)
  
**Does the point \( K(-2, 7, 10) \) belong to plane \( P \)? Justify.**

To check if \( K \) lies on the plane, verify if the vector \( \overrightarrow{AK} \) can be written as a linear combination of \( \overrightarrow{u} \) and \( \overrightarrow{v} \). This involves solving a system of equations.

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**Exercice 3:**

You are given two lines and two planes:

- Line \( d \) passing through \( A(1, 2, 0) \) with direction vector \( \overrightarrow{u}(-3, -1, 4) \)
- Plane \( P_1 \) passing through \( B(-1, 0, 3) \) with two direction vectors \( \overrightarrow{v}(-5, -2, 4) \) and \( \overrightarrow{w}(-4, 3, 40) \)
- Plane \( P_2 \) passing through \( C(0, 1, 2) \) with two direction vectors \( \overrightarrow{k}(0, 2, 3) \) and \( \overrightarrow{m}(1, 2, 0) \)

**1) Are line \( d \) and plane \( P_1 \) secant? Justify by detailing the calculations.**

To check if the line and the plane are secant, solve the parametric equations of the line and substitute them into the equation of the plane.

**2) Are the planes \( P_1 \) and \( P_2 \) parallel? Justify by detailing the calculations.**

For two planes to be parallel, their normal vectors must be proportional. You can find the normal vectors of the planes by taking the cross product of the direction vectors in each plane and checking if they are proportional.

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Let me know which part you would like to start with, or if you would like more detailed explanations for any specific problem!

ABCD is a tetrahedron. The midpoint I of [AD], and the points G and H are defined by: \( \overrightarrow{BG} = \overrightarrow{AB} \) and \( \overrightarrow{IH} = \frac{3}{2} \overrightarrow{AD} \). Find the following: 1. Express \( \overrightarrow{GC} \) as a linear combination of \( \overrightarrow{AB}, \overrightarrow{AC}, \overrightarrow{AD} \), 2. Express \( \overrightarrow{HC} \) as a linear combination, 3. Decompose \( \overrightarrow{GH} \) in the basis \( \overrightarrow{AB}, \overrightarrow{AC}, \overrightarrow{AD} \), etc.

Solve the advanced geometric vector problem involving a tetrahedron, including decomposing vectors and expressing them as linear combinations of other vectors. The problem covers vector algebra, decomposition in a vector space, and more, ideal for Grades 11-12.

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