From the image you uploaded, the task is asking for the vector sum $\mathbf{a} + \mathbf{b}$ to be written as a column vector. Here’s the process to solve this:

1. Identify the vectors from the grid:

   • The vector $\mathbf{a}$ is represented by the pink arrow, which starts at the origin and ends at the point $(5, 4)$. This means that $\mathbf{a} = \begin{pmatrix} 5 \\ 4 \end{pmatrix}$.
   • The vector $\mathbf{b}$ is represented by the green arrow. From the image, it starts at the origin and ends at the point $(3, -2)$. This means that $\mathbf{b} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$.

2. Add the vectors: To find $\mathbf{a} + \mathbf{b}$, add the corresponding components: $\mathbf{a} + \mathbf{b} = \begin{pmatrix} 5 \\ 4 \end{pmatrix} + \begin{pmatrix} 3 \\ -2 \end{pmatrix} = \begin{pmatrix} 5 + 3 \\ 4 + (-2) \end{pmatrix} = \begin{pmatrix} 8 \\ 2 \end{pmatrix}$

Thus, the sum of the vectors as a column vector is: $\mathbf{a} + \mathbf{b} = \begin{pmatrix} 8 \\ 2 \end{pmatrix}$

Would you like more details on how to perform vector addition or any further explanations?

Here are some additional questions that might help you deepen your understanding:

1. How do you represent vectors graphically?
2. What are the properties of vector addition?
3. How do you calculate the magnitude of a vector?
4. What is the difference between a row vector and a column vector?
5. How can you express a vector in terms of unit vectors?

Tip: Always ensure that you add corresponding components when performing vector addition, as shown in the calculation above.