To solve this problem, we need to express the green vector $\mathbf{x}$ as a linear combination of the two blue vectors $\mathbf{a}$ and $\mathbf{b}$. The equation would take the form:

$\mathbf{x} = c_1 \mathbf{a} + c_2 \mathbf{b}$

Where $c_1$ and $c_2$ are scalars (coefficients) for the vectors $\mathbf{a}$ and $\mathbf{b}$ respectively.

Step-by-Step Approach:

1. Identify the vectors:

   • Vector $\mathbf{a}$ appears to point in the positive x-direction (right), and its coordinates are approximately $\mathbf{a} = (3, 0)$.
   • Vector $\mathbf{b}$ points downward and to the left, with coordinates $\mathbf{b} = (-1, -2)$.

2. Coordinates of $\mathbf{x}$:

   • The green vector $\mathbf{x}$ appears to point upward to the left, approximately at $\mathbf{x} = (-1, 4)$.

3. Set up the linear combination: We need to solve for $c_1$ and $c_2$ in the following system: $c_1 \mathbf{a} + c_2 \mathbf{b} = \mathbf{x}$ Substituting the values for the vectors: $c_1 (3, 0) + c_2 (-1, -2) = (-1, 4)$

4. Break it into components:

   • For the x-component: $3c_1 - c_2 = -1$
   • For the y-component: $-2c_2 = 4$

5. Solve for $c_2$: From the second equation: $-2c_2 = 4 \quad \Rightarrow \quad c_2 = -2$

6. Solve for $c_1$: Substitute $c_2 = -2$ into the first equation: $3c_1 - (-2) = -1 \quad \Rightarrow \quad 3c_1 + 2 = -1$ $3c_1 = -3 \quad \Rightarrow \quad c_1 = -1$

Final Answer:

The vector $\mathbf{x}$ can be written as: $\mathbf{x} = -1 \cdot \mathbf{a} + (-2) \cdot \mathbf{b}$ Or: $\mathbf{x} = -\mathbf{a} - 2\mathbf{b}$

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you verify a linear combination of vectors?
2. What is the geometric meaning of a linear combination of two vectors?
3. How can we express a vector in different bases?
4. How would you represent three vectors in 3D space as a linear combination?
5. What are the conditions for vectors to form a linear basis?

Tip: When working with linear combinations, always ensure the vectors are linearly independent (i.e., they don’t lie on the same line) to guarantee that the system has a unique solution.