1. **State the problem:** We are given points $A(-4,5)$ and $B(6,1)$, and the vector $\overrightarrow{BC} = \begin{pmatrix} -3 \\ -4 \end{pmatrix}$. We need to find: (a) The column vector $\overrightarrow{AB}$. (b) The coordinates of point $C$. 2. **Find $\overrightarrow{AB}$:** The vector from point $A$ to point $B$ is given by subtracting coordinates of $A$ from $B$: $$\overrightarrow{AB} = \begin{pmatrix} x_B - x_A \\ y_B - y_A \end{pmatrix} = \begin{pmatrix} 6 - (-4) \\ 1 - 5 \end{pmatrix} = \begin{pmatrix} 6 + 4 \\ -4 \end{pmatrix} = \begin{pmatrix} 10 \\ -4 \end{pmatrix}$$ 3. **Find coordinates of $C$:** Since $\overrightarrow{BC} = \begin{pmatrix} -3 \\ -4 \end{pmatrix}$, and $B$ is at $(6,1)$, we add the vector $\overrightarrow{BC}$ to $B$ to get $C$: $$C = B + \overrightarrow{BC} = \begin{pmatrix} 6 \\ 1 \end{pmatrix} + \begin{pmatrix} -3 \\ -4 \end{pmatrix} = \begin{pmatrix} 6 - 3 \\ 1 - 4 \end{pmatrix} = \begin{pmatrix} 3 \\ -3 \end{pmatrix}$$ **Final answers:** (a) $\overrightarrow{AB} = \begin{pmatrix} 10 \\ -4 \end{pmatrix}$ (b) Coordinates of $C$ are $(3, -3)$.