1. **State the problem:** We are given vectors \(p = \begin{pmatrix}1 \\ -3\end{pmatrix}\) and \(q = \begin{pmatrix}-2 \\ 0\end{pmatrix}\). (i) Find the magnitude \(|p|\). (ii) Find the vector \(p - q\). (iii) Given \(r = ap + bq\) and \(r = 4p + 6q\), find \(a\) and \(b\). 2. **Formula for magnitude of a vector:** \[|p| = \sqrt{p_1^2 + p_2^2}\] where \(p_1\) and \(p_2\) are components of \(p\). 3. **Calculate \(|p|\):** \[ |p| = \sqrt{1^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \] 4. **Calculate \(p - q\):** \[ p - q = \begin{pmatrix}1 \\ -3\end{pmatrix} - \begin{pmatrix}-2 \\ 0\end{pmatrix} = \begin{pmatrix}1 - (-2) \\ -3 - 0\end{pmatrix} = \begin{pmatrix}3 \\ -3\end{pmatrix} \] 5. **Find \(a\) and \(b\) given \(r = ap + bq\) and \(r = 4p + 6q\):** Since \(r = 4p + 6q\), by comparing, we have: \[ a = 4, \quad b = 6 \] **Final answers:** - \(|p| = \sqrt{10}\) - \(p - q = \begin{pmatrix}3 \\ -3\end{pmatrix}\) - \(a = 4\), \(b = 6\)