https://file.aiquickdraw.com/mathbot/file/1726554977795fjzfow4b.jpg

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!

Les vecteurs \(\vec{u}(a) = (2, a)\) et \(\vec{v}(2a) = (-1, -2a)\) sont colinéaires pour quelles valeurs de \(a\) ?

Learn how to determine the collinearity of vectors \(\vec{u}(a) = (2, a)\) and \(\vec{v}(2a) = (-1, -2a)\). This solution covers key concepts of vector algebra, including the condition for collinearity and solving for unknown variables.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation