Question 3 9 marks Paper 1 Section A Medium No calculator Free
Maximum mark: 9
The position vectors of the points A, B and C are \( \mathbf{a} \), \( \mathbf{b} \) and \( \mathbf{c} \) respectively, relative to an origin O. The following diagram shows the triangle ABC and points M, R, S and T.
M is the midpoint of [AC].
R is a point on [AB] such that \( \overrightarrow{AR} = \dfrac{1}{3}\overrightarrow{AB} \).
S is a point on [AC] such that \( \overrightarrow{AS} = \dfrac{2}{3}\overrightarrow{AC} \).
T is a point on [RS] such that \( \overrightarrow{RT} = \dfrac{2}{3}\overrightarrow{RS} \).
(a) (i) Express \( \overrightarrow{AM} \) in terms of \( \mathbf{a} \) and \( \mathbf{c} \).[1]
(ii) Hence show that \( \overrightarrow{BM} = \dfrac{1}{2}\mathbf{a} - \mathbf{b} + \dfrac{1}{2}\mathbf{c} \).[2]
(b) (i) Express \( \overrightarrow{RA} \) in terms of \( \mathbf{a} \) and \( \mathbf{b} \).[1]
(ii) Show that \( \overrightarrow{RT} = -\dfrac{2}{9}\mathbf{a} - \dfrac{2}{9}\mathbf{b} + \dfrac{4}{9}\mathbf{c} \).[3]
(c) Prove that T lies on [BM].[2]
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