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?

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.