Sure! Let's break down the image and solve the exercises step by step. Here's an overview of the tasks:

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Exercise 1

You are given three points $A(1; -1)$, $B(4; 7)$, and $C(2; 0)$. The question asks: "Les points A, B, C sont-ils alignés ?" This requires determining if the three points are collinear.

To solve:

• Calculate the vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$.
• Check if $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are collinear by verifying if one is a scalar multiple of the other.

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Exercise 2

You need to:

1. Discuss the collinearity of two vectors $\vec{u}$ and $\vec{v}$ for given cases of $m$ (the real number parameter).
2. Cases involve:
   • $\vec{u} = m \vec{i} + 3 \vec{j}$ and $\vec{v} = 2 \vec{i} + \vec{j}$
   • $\vec{u} = (m + 3; m - 3)$ and $\vec{v} = (8; m - 3)$
   • $\vec{u} = (-2m; m - 2)$ and $\vec{v} = (2m + 5; -m + 1)$

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Exercise 3

1. Prove that $\overrightarrow{OB}$ and $\overrightarrow{OF}$ are collinear, where:
   • $\overrightarrow{OF} = \frac{1}{5} \overrightarrow{OC}$ and $\overrightarrow{OE} = -\frac{1}{5} \overrightarrow{OC}$.
2. Determine the equations of lines passing through points.

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I'll solve these step by step, starting with Exercise 1.

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Exercise 1 Solution

Step 1: Calculate vectors

• $\overrightarrow{AB} = (4 - 1; 7 - (-1)) = (3; 8)$
• $\overrightarrow{AC} = (2 - 1; 0 - (-1)) = (1; 1)$

Step 2: Check collinearity

Vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are collinear if: $\frac{x_1}{x_2} = \frac{y_1}{y_2}$ Substitute values: $\frac{3}{1} \neq \frac{8}{1}$

Since the proportions are not equal, the points A, B, and C are not collinear.

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Would you like me to proceed with Exercise 2? Let me know!