1. **State the problem:** Given vectors $a = \begin{pmatrix} -2 \\ 3 \end{pmatrix}$ and $b = \begin{pmatrix} 5 \\ -1 \end{pmatrix}$, find the column vectors for: (i) $a + b$ (ii) $2a - b$ 2. **Recall vector addition and scalar multiplication:** - To add vectors, add their corresponding components. - To multiply a vector by a scalar, multiply each component by that scalar. 3. **Calculate (i) $a + b$:** $$a + b = \begin{pmatrix} -2 \\ 3 \end{pmatrix} + \begin{pmatrix} 5 \\ -1 \end{pmatrix} = \begin{pmatrix} -2 + 5 \\ 3 + (-1) \end{pmatrix} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$$ 4. **Calculate (ii) $2a - b$:** First, calculate $2a$: $$2a = 2 \times \begin{pmatrix} -2 \\ 3 \end{pmatrix} = \begin{pmatrix} 2 \times -2 \\ 2 \times 3 \end{pmatrix} = \begin{pmatrix} -4 \\ 6 \end{pmatrix}$$ Then subtract $b$: $$2a - b = \begin{pmatrix} -4 \\ 6 \end{pmatrix} - \begin{pmatrix} 5 \\ -1 \end{pmatrix} = \begin{pmatrix} -4 - 5 \\ 6 - (-1) \end{pmatrix} = \begin{pmatrix} -9 \\ 7 \end{pmatrix}$$ **Final answers:** (i) $a + b = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$ (ii) $2a - b = \begin{pmatrix} -9 \\ 7 \end{pmatrix}$