The question asks for which values of $a$ the vectors $\vec{u}(a) = (2, a)$ and $\vec{v}(2a) = (-1, -2a)$ are collinear.

Collinearity Condition:

Two vectors $\vec{u} = (u_1, u_2)$ and $\vec{v} = (v_1, v_2)$ are collinear if there exists a scalar $k$ such that: $\vec{u} = k \vec{v}$ This leads to the component-wise equation: $\frac{u_1}{v_1} = \frac{u_2}{v_2} = k$ Using this on the given vectors: $\frac{2}{-1} = \frac{a}{-2a}$ We can now solve this equation to determine the value of $a$.

Let me calculate this for you.The equation $\frac{2}{-1} = \frac{a}{-2a}$ has no solution for $a$. This means there is no value of $a$ that makes the vectors $\vec{u}(a) = (2, a)$ and $\vec{v}(2a) = (-1, -2a)$ collinear.

Thus, the correct answer is option d) aucune valeur de a.

Do you have any further questions or would you like more details? Here are 5 related questions for practice:

1. What is the general condition for two vectors to be orthogonal?
2. How would the calculation change if the vectors were $\vec{u}(a) = (a, 2)$ and $\vec{v}(2a) = (1, 2a)$?
3. Can two vectors of different dimensions be collinear? Why or why not?
4. How do you check if three vectors are coplanar?
5. What role does the scalar multiple $k$ play in vector operations?

Tip: When determining vector collinearity, always remember to compare the components through ratios or the cross-product for 3D vectors!