https://file.aiquickdraw.com/mathbot/file/1727202427037452lc0a1.jpg

Let's go through the two exercises and their respective tasks:

### Exercise 1

1. **Vectors:**
   \[
   \vec{u} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}, \quad \vec{v} = \begin{pmatrix} 6 \\ 2 \end{pmatrix}
   \]
   You are asked to:
   - Show that \(\vec{u}\) and \(\vec{v}\) are **not collinear**.
   - Express the vector \(\vec{w} = \begin{pmatrix} 4 \\ 3 \end{pmatrix}\) as a **linear combination** of \(\vec{u}\) and \(\vec{v}\).
   - Graphically represent the vectors \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\).

#### Solution Outline:
- **Non-collinearity**: Two vectors are collinear if one is a scalar multiple of the other. Here, check if:
  \[
  \vec{v} = k\vec{u}
  \]
  for some scalar \(k\). If no such \(k\) exists, then the vectors are not collinear.
  
- **Linear Combination**: The vector \(\vec{w}\) can be written as a linear combination of \(\vec{u}\) and \(\vec{v}\), i.e.,
  \[
  \vec{w} = a\vec{u} + b\vec{v}
  \]
  Solve for \(a\) and \(b\) by solving the system of equations obtained from comparing components.

### Exercise 2

2. **Vectors:**
   \[
   \vec{u} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}, \quad \vec{v} = \begin{pmatrix} 6 \\ 3 \end{pmatrix}
   \]
   You are asked to:
   - Show that \(\vec{u}\) and \(\vec{v}\) are **collinear**.
   - Determine if it's possible to express the vector \(\vec{w} = \begin{pmatrix} 4 \\ 3 \end{pmatrix}\) as a **linear combination** of \(\vec{u}\) and \(\vec{v}\).
   - Graphically represent the vectors \(\vec{u}\), \(\vec{v}\), and \(\vec{w}\), and describe the set of vectors that can be expressed as a linear combination of \(\vec{u}\) and \(\vec{v}\).

#### Solution Outline:
- **Collinearity**: Verify if there is a scalar \(k\) such that:
  \[
  \vec{v} = k\vec{u}
  \]
  If \(k\) exists, the vectors are collinear.

- **Linear Combination**: As in Exercise 1, express \(\vec{w}\) as \(a\vec{u} + b\vec{v}\) and solve for \(a\) and \(b\) if possible.

Would you like the detailed calculations or graphical representations for any part? 

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### Related Questions:
1. What does it mean geometrically when two vectors are collinear?
2. How do you solve a system of equations to find a linear combination?
3. What is the geometric interpretation of expressing one vector as a linear combination of others?
4. How can you determine if three or more vectors form a basis in the plane?
5. How does the concept of vector independence apply to these problems?

**Tip**: When solving for linear combinations, always check if the resulting system of equations is consistent to avoid errors.

1. Soient \( \vec{u} = \begin{pmatrix} 2 \\ 1 \end{pmatrix} \) et \( \vec{v} = \begin{pmatrix} 6 \\ 2 \end{pmatrix} \). Montrer que \( \vec{u} \) et \( \vec{v} \) ne sont pas colinéaires. Exprimer le vecteur \( \vec{w} = \begin{pmatrix} 4 \\ 3 \end{pmatrix} \) en tant que combinaison linéaire de \( \vec{u} \) et \( \vec{v} \). Représenter graphiquement les vecteurs \( \vec{u}, \vec{v} \) et \( \vec{w} \). 

2. Soient \( \vec{u} = \begin{pmatrix} 2 \\ 1 \end{pmatrix} \) et \( \vec{v} = \begin{pmatrix} 6 \\ 3 \end{pmatrix} \). Montrer que \( \vec{u} \) et \( \vec{v} \) sont colinéaires. Est-il possible d'exprimer le vecteur \( \vec{w} = \begin{pmatrix} 4 \\ 3 \end{pmatrix} \) en tant que combinaison linéaire de \( \vec{u} \) et \( \vec{v} \)?

Explore vector algebra by proving the non-collinearity of vectors, expressing a vector as a linear combination, and demonstrating the collinearity of two vectors. Includes graphical representations and step-by-step solutions.

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