1. **State the problem:** We are given vectors $\mathbf{a} = 2\mathbf{i} + 5\mathbf{j}$, $\mathbf{b} = 12\mathbf{i} - 10\mathbf{j}$, and $\mathbf{c} = -3\mathbf{i} + 9\mathbf{j}$. We need to find the vector expression $\mathbf{a} + 2\mathbf{b} - \mathbf{c}$ using column matrix notation. 2. **Recall vector addition and scalar multiplication:** - Vector addition is done component-wise: if $\mathbf{u} = \begin{bmatrix} u_x \\ u_y \end{bmatrix}$ and $\mathbf{v} = \begin{bmatrix} v_x \\ v_y \end{bmatrix}$, then $\mathbf{u} + \mathbf{v} = \begin{bmatrix} u_x + v_x \\ u_y + v_y \end{bmatrix}$. - Scalar multiplication multiplies each component by the scalar: $k\mathbf{u} = \begin{bmatrix} k u_x \\ k u_y \end{bmatrix}$. 3. **Write vectors in column matrix form:** $$ \mathbf{a} = \begin{bmatrix} 2 \\ 5 \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 12 \\ -10 \end{bmatrix}, \quad \mathbf{c} = \begin{bmatrix} -3 \\ 9 \end{bmatrix} $$ 4. **Calculate $2\mathbf{b}$:** $$ 2\mathbf{b} = 2 \times \begin{bmatrix} 12 \\ -10 \end{bmatrix} = \begin{bmatrix} 24 \\ -20 \end{bmatrix} $$ 5. **Calculate $\mathbf{a} + 2\mathbf{b}$:** $$ \begin{bmatrix} 2 \\ 5 \end{bmatrix} + \begin{bmatrix} 24 \\ -20 \end{bmatrix} = \begin{bmatrix} 2 + 24 \\ 5 + (-20) \end{bmatrix} = \begin{bmatrix} 26 \\ -15 \end{bmatrix} $$ 6. **Calculate $\mathbf{a} + 2\mathbf{b} - \mathbf{c}$:** $$ \begin{bmatrix} 26 \\ -15 \end{bmatrix} - \begin{bmatrix} -3 \\ 9 \end{bmatrix} = \begin{bmatrix} 26 - (-3) \\ -15 - 9 \end{bmatrix} = \begin{bmatrix} 29 \\ -24 \end{bmatrix} $$ 7. **Final answer:** The vector $\mathbf{a} + 2\mathbf{b} - \mathbf{c}$ in column matrix notation is $$ \boxed{\begin{bmatrix} 29 \\ -24 \end{bmatrix}} $$ This means the resulting vector points 29 units in the $\mathbf{i}$ (x) direction and -24 units in the $\mathbf{j}$ (y) direction.