The problem presented is about the criterion of collinearity of two vectors, $\vec{a}$ and $\vec{b}$, in a given basis.

To determine collinearity, two vectors $\vec{a}$ and $\vec{b}$ are collinear if there exists a real number $\alpha$ such that: $\vec{b} = \alpha \vec{a}$ This means that $\vec{b}$ can be represented as a scalar multiple of $\vec{a}$, which is the classic criterion for collinearity.

Examining the options:

1. Option 1 suggests a condition involving $\vec{b} + \alpha = \alpha \vec{a}$, which is incorrect.
2. Option 2 correctly states that $\vec{b} = \alpha \vec{a}$.
3. Option 3 states that the vectors are collinear if they are defined in the same basis, which is not a sufficient condition for collinearity.
4. Option 4 is similar to Option 1, which is also incorrect due to the incorrect formulation.

Thus, the correct answer is Option 2, as it accurately describes the criterion for collinearity.

Would you like further details or have any other questions?

Here are five related questions to expand your understanding:

1. What is the geometric interpretation of collinear vectors in 3D space?
2. How does collinearity change when considering vectors in different bases?
3. What are examples of non-collinear vectors in 2D and 3D space?
4. How is collinearity used in determining the properties of shapes and lines in geometry?
5. What is the relationship between collinearity and vector cross products?

Tip: Collinearity can often be visually verified by checking if vectors lie along the same line or direction, even in different magnitudes.