0606 P12 - Nov 2021 - Q7 - 8 marks
8026
(a) The diagram shows triangle \(OAC\), where \(\overrightarrow{OA}=\mathbf a\), \(\overrightarrow{OB}=\mathbf b\) and \(\overrightarrow{OC}=\mathbf c\). The point \(B\) lies on the line \(AC\) such that \(AB:BC=m:n\), where \(m\) and \(n\) are constants.
(i) Write down \(\overrightarrow{AB}\) in terms of \(\mathbf a\) and \(\mathbf b\).
(ii) Write down \(\overrightarrow{BC}\) in terms of \(\mathbf b\) and \(\mathbf c\).
(iii) Hence show that \(n\mathbf a+m\mathbf c=(m+n)\mathbf b\).
(b) Given that
\(\lambda\begin{pmatrix}2\\1\end{pmatrix}+(\mu-1)\begin{pmatrix}-4\\7\end{pmatrix} =(\lambda+1)\begin{pmatrix}4\\-2\end{pmatrix},\)
find the value of each of the constants \(\lambda\) and \(\mu\).
