To express the vector in the bottom grid as a combination of vectors $\mathbf{u}$ and $\mathbf{v}$ shown in the top grid, we need to determine how many steps in each direction are required.

Step-by-Step Solution

1. Identify the Vector Directions:

   • Vector $\mathbf{u}$ points to the right and slightly upward (along the positive x-axis).
   • Vector $\mathbf{v}$ points to the left and upward.

2. Determine the Position of the Target Vector:

   • The vector in the lower diagram points to the right and slightly upward, similar to vector $\mathbf{u}$, but longer.
   • To reach the vector in the lower diagram, we may need multiple steps of $\mathbf{u}$ and possibly one of $\mathbf{v}$ to adjust its direction.

3. Calculate the Combination:

   • From visual inspection, it appears the target vector can be reached by taking 3 steps of $\mathbf{u}$ and 1 step of $\mathbf{v}$.
   • Therefore, the vector can be written as: $\text{Vector} = 3 \mathbf{u} + 1 \mathbf{v}$

Answer

$\text{Vector} = 3 \mathbf{u} + \mathbf{v}$

Would you like further details on how to calculate vector combinations or verify components?

1. How can we find the exact coordinates of vectors $\mathbf{u}$ and $\mathbf{v}$?
2. What if the target vector was in a different quadrant? How would that change the combination?
3. How can we generalize this approach for any given pair of vectors?
4. What is the geometric interpretation of vector addition?
5. How can we verify the correctness of this vector combination algebraically?

Tip: Visualizing vector addition as "head-to-tail" helps in understanding how each vector contributes to the final position.