Specialist Mathematics · Unit 1 · Vectors in the plane · Representing vectors in the plane by directed line segments
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In the diagram below, $OABC$ is a parallelogram with $\vec{OA} = \mathbf{a}$ and $\vec{OC} = \mathbf{c}$. Point $P$ lies on $AC$ such that $AP:PC = 2:1$, and point $Q$ is the midpoint of $OB$. (a) Express $\vec{OP}$ in terms of $\mathbf{a}$ and $\mathbf{c}$. (b) Express $\vec{OQ}$ in terms of $\mathbf{a}$ and $\mathbf{c}$. (c) Hence determine $\vec{PQ}$ in terms of $\mathbf{a}$ and $\mathbf{c}$.
Three hikers are positioned at points P, Q, and R on a flat plain. Vector $\vec{PQ}$ is represented by $\mathbf{a}$ and vector $\vec{PR}$ is represented by $\mathbf{b}$. Point M lies on segment QR such that M divides QR in the ratio 2:1, with M closer to R. (a) Express $\vec{QR}$ in terms of $\mathbf{a}$ and $\mathbf{b}$. (1 mark) (b) Hence, express $\vec{PM}$ in terms of $\mathbf{a}$ and $\mathbf{b}$. (2 marks)
Two vectors **a** and **b** are shown on the diagram below. Which of the following represents the vector **c** in terms of **a** and **b**?