Given that \(p = \begin{pmatrix} 5 \\ 3 \end{pmatrix}, q = \begin{pmatrix} -1 \\ 2 \end{pmatrix}\) and \(r = \begin{pmatrix} 17 \\ 5 \end{pmatrix}\) and \(r...
Question 1 Report
Given that \(p = \begin{pmatrix} 5 \\ 3 \end{pmatrix}, q = \begin{pmatrix} -1 \\ 2 \end{pmatrix}\) and \(r = \begin{pmatrix} 17 \\ 5 \end{pmatrix}\) and \(r = \alpha r + \beta q\), where \(\alpha\) and \(\beta\) are scalars, express q in terms of r and p.
Answer Details
Express \(r\) as a linear combination \(r = \alpha p + \beta q\):
\[\begin{pmatrix} 17 \\ 5 \end{pmatrix} = \alpha\begin{pmatrix} 5 \\ 3 \end{pmatrix} + \beta\begin{pmatrix} -1 \\ 2 \end{pmatrix}\]This gives the equations:
\[5\alpha - \beta = 17,\qquad 3\alpha + 2\beta = 5\]From the first, \(\beta = 5\alpha - 17\). Substitute:
\[3\alpha + 2(5\alpha - 17) = 5 \ \Rightarrow\ 13\alpha = 39 \ \Rightarrow\ \alpha = 3,\quad \beta = -2\]Hence \(r = 3p - 2q\). Make \(q\) the subject:
\[2q = 3p - r \ \Rightarrow\ q = \tfrac{1}{2}(3p - r)\]Check: \(\tfrac{1}{2}\left(3\begin{pmatrix}5\\3\end{pmatrix} - \begin{pmatrix}17\\5\end{pmatrix}\right) = \tfrac{1}{2}\begin{pmatrix}-2\\4\end{pmatrix} = \begin{pmatrix}-1\\2\end{pmatrix} = q\).
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