Choosing a point to make three points collinear
Choosing a point to make three points collinear
IP 4
8 marks
In triangle \(OAB\), \(\overrightarrow{OA}=\mathbf a\) and \(\overrightarrow{OB}=\mathbf b\). The point \(P\) lies on \(OA\) with \(OP:PA=3:1\), and \(Q\) is the midpoint of \(AB\). The point \(R\) lies on \(OB\) produced, with \(\overrightarrow{OR}=k\mathbf b\).
- Express \(\overrightarrow{PQ}\) in terms of \(\mathbf a\) and \(\mathbf b\).[2]
- Given that \(P\), \(Q\) and \(R\) lie on a straight line, find the value of \(k\). State clearly the property of \(\mathbf a\) and \(\mathbf b\) that you use.[4]
- Find the ratio \(PQ:QR\).[2]
Solution:
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Answer:(a) \(-\dfrac14\mathbf a+\dfrac12\mathbf b\) (b) \(k=\dfrac32\), using that \(\mathbf a\) and \(\mathbf b\) are non-zero and non-parallel (c) \(1:2\)
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