Vector
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$(\mathrm{CIE} 0606 / 2018 / \mathrm{w} / 11 / \mathrm{q} 10)$ Particle $A$ is at the point with position vector $\left(\begin{array}{r}2 \\ -5\end{array}\right)$ at time $t=0$ and moves with a speed of $10 \mathrm{~ms}^{-1}$ in the same direction as $\left(\begin{array}{l}3 \\ 4\end{array}\right)$ (i) Given that $A$ is at the point with position vector $\left(\begin{array}{c}38 \\ a\end{array}\right)$ when $t=6 \mathrm{~s}$, find the value of the constant $a$. [3] Particle $B$ is at the point with position vector $\left(\begin{array}{l}16 \\ 37\end{array}\right)$ at time $t=0$ and moves with velocity $\left(\begin{array}{l}4 \\ 2\end{array}\right) \mathrm{ms}^{-1}$ (ii) Write down, in terms of $t$, the position vector of $B$ at time $t \mathrm{~s}$$[1]$ (iii) Verify that particles $A$ and $B$ collide. (iv) Write down the position vector of the point of collision.$[1]$ |
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$(\mathrm{CIE} 0606 / 2018 / \mathrm{w} / 12 / \mathrm{q} 7)$ (a) The vector $v$ has a magnitude of 39 units and is in the same direction as $\left(\begin{array}{r}-12 \\ 5\end{array}\right)$. Write $\mathrm{v}$ in the form $\left(\begin{array}{l}a \\ b\end{array}\right)$, where $a$ and $b$ are constants. (b) Vectors $\mathbf{p}$ and $q$ are such that $p=\left(\begin{array}{c}r+s \\ r+6\end{array}\right)$ and $q=\left(\begin{array}{c}5 r+1 \\ 2 s-1\end{array}\right)$, where $r$ and $s$ are constants. Given that $2 \mathbf{p}+3 \mathbf{q}=\left(\begin{array}{l}0 \\ 0\end{array}\right)$, find the value of $r$ and of $s$ |
| 3 | (CIE $0606 / 2018 / \mathrm{w} / 13 / \mathrm{q} 7)$ The diagram shows a quadrilateral $O A B C .$ The point $D$ lies on $O B$ such that $\overrightarrow{O D}=2 \overrightarrow{D B}$ and $\overrightarrow{A D}=m \overrightarrow{A C}$, where $m$ is a scalar quantity. $$\overrightarrow{O A}=\mathrm{a} \quad \overrightarrow{O B}=\mathbf{b} \quad \overrightarrow{O C}=\mathrm{c}$$ (i) Find $\overrightarrow{A D}$ in terms of $m$, a and $\mathbf{c}$.$[1]$ (ii) Find $\overrightarrow{A D}$ in terms of a and $\mathbf{b}$.$[2]$ (iii) Given that $15 \mathrm{a}=16 \mathrm{~b}-9 \mathrm{c}$, find the value of $m$. |
| 4 | $(\mathrm{CIE} 0606 / 2019 / \mathrm{s} / 12 / \mathrm{q} 7)$ A pilot wishes to fly his plane from a point $A$ to a point $B$ on a bearing of $055^{\circ}$. There is a wind blowing at $120 \mathrm{~km} \mathrm{~h}^{-1}$ from the west. The plane can fly at $650 \mathrm{~km} \mathrm{~h}^{-1}$ in still air. (i) Find the direction in which the pilot must fly his plane in order to reach $B$. (ii) Given that the distance between $A$ and $B$ is $1250 \mathrm{~km}$, find the time it will take the pilot to fly from $A$ to $B$$[4]$ |
| 5 | $(\mathrm{CIE} 0606 / 2019 / \mathrm{s} / 13 / \mathrm{q} 11)$ A pilot wishes to fly his plane from a point $A$ to a point $B$. The bearing of $B$ from $A$ is $050^{\circ}$. A wind is blowing from the north at a speed of $120 \mathrm{~km} \mathrm{~h}^{-1}$. The plane can fly at $600 \mathrm{~km} \mathrm{~h}^{-1}$ in still air. (i) Find the bearing on which the pilot must fly his plane in order to reach $B$. (ii) Given that the distance from $A$ to $B$ is $2500 \mathrm{~km}$, find the time taken to fly from $A$ to $B$.[4] |
| 6 | $(\mathrm{CIE} 0606 / 2019 / \mathrm{s} / 22 / \mathrm{q} 10)$ (a) Find the unit vector in the direction of $5 \mathbf{i}-15 \mathbf{j}$. (b) The position vectors of points $A$ and $B$ relative to an origin $O$ are $\left(\begin{array}{r}3 \\ -5\end{array}\right)$ and $\left(\begin{array}{c}12 \\ 7\end{array}\right)$ respectively. The point $C$ lies on $A B$ such that $A C: C B$ is $2: 1$. (i) Find the position vector of $C$ relative to $O$. The point $D$ lies on $O B$ such that $O D: O B$ is $1: \lambda$ and $\overrightarrow{D C}=\left(\begin{array}{c}6 \\ 1.25\end{array}\right)$ (ii) Find the value of $\lambda$. |
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$(\mathrm{CIE} 0606 / 2019 / \mathrm{s} / 23 / \mathrm{q} 10)$ Solution  The diagram shows a triangle $O A B$. The point $P$ is the midpoint of $O A$ and the point $Q$ lies on $O B$ such that $\overrightarrow{O Q}=\frac{1}{4} \overrightarrow{O B}$. The position vectors of $P$ and $Q$ relative to $O$ are $\mathrm{p}$ and $\mathrm{q}$ respectively. (i) Find, in terms of $\mathbf{p}$ and $\mathrm{q}$, an expression for each of the vectors $\overrightarrow{P Q}, \overrightarrow{Q A}$ and $\overrightarrow{P B}$. (ii) Given that $\overrightarrow{P R}=\lambda \overrightarrow{P B}$ and that $\overrightarrow{Q R}=\mu \overrightarrow{Q A}$, find an expression for $\overrightarrow{P Q}$ in terms of $\lambda, \mu, \mathbf{p}$ and $\mathbf{q}$. (iii) Using your expressions for $\overrightarrow{P Q}$, find the value of $\lambda$ and of $\mu$.$[4]$ |
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$(CIE $0606 / 2020 / \mathrm{w} / 13 / \mathrm{q} 9)$  The diagram shows the triangle $O A C$. The point $B$ is the midpoint of $O C$. The point $Y$ lies on $A C$ such that $O Y$ intersects $A B$ at the point $X$ where $A X: X B=3: 1$. It is given that $\overrightarrow{O A}=\mathbf{a}$ and $\overrightarrow{O B}=\mathbf{b}$. (a) Find $\overrightarrow{O X}$ in terms of a and $\mathbf{b}$, giving your answer in its simplest form.$[3]$ (b) Find $\overrightarrow{A C}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.$[1]$ (c) Given that $\overrightarrow{O Y}=h \overrightarrow{O X}$, find $\overrightarrow{A Y}$ in terms of $\mathbf{a}, \mathbf{b}$ and $h .$$[1]$ (d) Given that $\overrightarrow{A Y}=m \overrightarrow{A C}$, find the value of $h$ and of $m$.$[4]$ |
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$(\mathrm{CIE} 0606 / 2020 / \mathrm{w} / 22 / \mathrm{q} 9)$  In the diagram $\overrightarrow{O P}=2 \mathbf{b}, \overrightarrow{O S}=3 \mathbf{a}, \overrightarrow{S R}=\mathbf{b}$ and $\overrightarrow{P Q}=\mathbf{a}$. The lines $O R$ and $Q S$ intersect at $X$. (a) Find $\overrightarrow{O Q}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.$[1]$ (b) Find $\overrightarrow{Q S}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.$[1]$ (c) Given that $\overrightarrow{Q X}=\mu \overrightarrow{Q S}$, find $\overrightarrow{O X}$ in terms of $\mathbf{a}, \mathbf{b}$ and $\mu$.$[1]$ (d) Given that $\overrightarrow{O X}=\lambda \overrightarrow{O R}$, find $\overrightarrow{O X}$ in terms of $\mathrm{a}, \mathbf{b}$ and $\lambda$.$[1]$ (e) Find the value of $\lambda$ and of $\mu$.$[3]$ (f) Find the value of $\frac{Q X}{X S}$. (g) Find the value of $\frac{O R}{O X}$.$[1]$ |
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$( 2018/$m/12/q6a$) $
 (ii) Find $\overrightarrow{OM}$ in terms of a and c.[2] (iii) Find $\overrightarrow{OM}$ in terms of b. $\left[1\right]$ $\mathrm{( iv) }$ Find $5\mathbf{a}+10\mathbf{c}$ in terms of b. [2] $\mathbf{(v)}$ Find $\overrightarrow{AB}$ in terms of a and c, giving your answer in its simplest form. [2] |
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(CIE/0606/2018/m/22/q5)
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$( 2018/$s$/11/$q8a$) $ $\mathbf{(a)}$ Given that $\mathbf{p=2i-5j$ and $q=i-3j}$, find the unit vector in the direction of 3p $-4\mathbf{q}.$ [4] (b)  A river flows between parallel banks at a speed of 1.25kmh$^{-1}$.A boy standing at point $A$ on one bank sends a toy boat across the river to his father standing directly opposite at point $B$. The toy boat, which can travel at $\nu$kmh$^{-1}$ in still water, crosses the river with resultant speed 2.73 kmh$^{-1}$ along the line $AB.$ (i) Calculate the value of v. [2] The direction in which the boy points the boat makes an angle $\theta$ with the line $AB.$ $( \mathbf{ii}) \quad $ Find the value of $\theta.$ [2] |
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