Question Bank | Unit 6B

1. In the diagram in the answer space $\vec{A B}$ represents the column vector $(\begin{matrix} 4 \\ - 1 \end{matrix})$ and $\vec{B C}$ represents the column vector $(\begin{matrix} - 2 \\ - 3 \end{matrix})$ .

(a) Mark clearly on the diagram, and label, the point $D$ such that $A B C D$ is a parallelogram.

(b) Write down $\vec{A D}$ in column vector form.

(c) Write down $\vec{C A}$ in column vector form.

2. (a) In the diagram, $T$ is the mid-point of $\vec{A B}$ and $M$ is the mid-point of $A T$ . Given that $\vec{O A} = a$ and $\vec{O B} = b$ , express as simply as possible in terms of $a$ and $b$ ,

(i) $\vec{A B}$ ,

(ii) $\vec{A M}$ ,

(iii) $\vec{O M}$ .

(b) Two points $P$ and $Q$ have position vectors $p$ and $q$ respectively, relative to the origin $O$ . Given that $p = (\begin{matrix} 5 \\ 3 \end{matrix})$ and $\vec{P Q} = (\begin{matrix} - 2 \\ 1 \end{matrix})$ , find

(i) $q$ ,

(ii) $| \vec{P Q} |$ ,

(iii) the coordinates of the point $R$ , which is such that $\vec{O R} = \vec{Q P}$ . Given also that $s = (\begin{matrix} 1 \\ 1 \end{matrix})$ , $t = (\begin{matrix} 8 \\ 2 \end{matrix})$ and $l p + m s = t$ , write down two simultaneous equations in $l$ and $m$ , and solve them.

3. Given that $\vec{O A} = (\begin{matrix} p \\ 4 \end{matrix})$ and $\vec{O B} = (\begin{matrix} 5 \\ 0 \end{matrix})$ , find

(a) the value of $| \vec{OB} |$ ,

(b) a value for $p$ if $\vec{O A}$ and $\vec{O B}$ are two sides of a rhombus.

4. $O A B C$ is a parallelogram where $O$ is the origin, $\vec{O A} = (\begin{matrix} 1 \\ 4 \end{matrix})$ and $\vec{O C} = (\begin{matrix} 5 \\ 2 \end{matrix})$ .

(a) On the diagram in the answer column mark and clearly label the points $A$ , $B$ , and $C$ .

(b) Express as a column vector

(i) $\vec{O B}$ ,

(ii) $\vec{C A}$ .

5. In the diagram $\vec{O P} = a$ and $\vec{O S} = b$ .

(a) Express $\vec{S P}$ in terms of $a$ and $b$ .

(b) Given that $\vec{S X} = h \vec{S P}$ , show that $\vec{O X} = h a + (1 - h) b$ .

(c) Given that $\vec{O Q} = 3 a$ and $\vec{Q R} = 2 b$ , write down an expression for $\vec{O R}$ in terms of $a$ and $b$ .

(d) Given that $\vec{O X} = k \vec{O R}$ use the results of parts (b) and (c) to find the values of $h$ and $k$ .

(e) Find the numerical value of the ratio $\frac{P X}{X S}$ .

6. It is given that $u = 4 a + 3 b$ , $v = 5 a - b$ and $w = h a + (h + k) b$ , where $h$ and $k$ are constants. If $w = 3 u - 2 v$ , calculate the value of $h$ and of $k$ .

7. In the diagram, $M$ and $N$ are the midpoints of $A D$ and $D C$ respectively. It is given that $\vec{A B} = \vec{B C} = a$ and that $\vec{B D} = a + b$ . Write down, as simply as possible in terms of $a$ and/or $b$ , expressions for

(a) $\vec{A D}$ ,

(b) $\vec{D C}$ ,

(c) $\vec{M N}$ .

8. (a) Given that $\vec{O K} = (\begin{matrix} 16 \\ 2 \end{matrix})$ , $\vec{O L} = (\begin{matrix} 4 \\ - 3 \end{matrix})$ and that $M$ and $N$ are the midpoints of $O K$ and $O L$ respectively,

(i) express $\vec{M N}$ as a column vector,

(ii) find the value of $| \vec{KL} |$ .

(b) Given that $\vec{O A} = 4 a$ , $\vec{O B} = 4 b$ and $\vec{B P} = 3 a - b$ , express as simply as possible, in terms of $a$ and $b$ ,

(i) $\vec{O P}$ ,

(ii) $\vec{A P}$ .

The lines $\vec{O A}$ produced and $\vec{B P}$ produced meet at $Q$ . Given that $\vec{B Q} = m \vec{B P}$ and $\vec{O Q} = n \vec{O A}$ , form an equation connecting $m$ , $n$ , $a$ and $b$ .

9. (a) The vectors $r$ and $s$ are defined by $r = (\begin{matrix} 2 \\ 4 \end{matrix})$ and $s = (\begin{matrix} - 1 \\ 3 \end{matrix})$ .

(i) Evaluate $| r |$ .

(ii) If $r + 2 t = 6 s$ , express $t$ as a column vector.

(b) In the triangle $A B C$ , $X$ is the midpoint of $A B$ .

(i) Given that $\vec{A B} = p$ and $\vec{A C} = q$ , express the following vectors in terms of $p$ and/or $q$ , giving your answers in their simplest form.

(I) $\vec{A X}$ ,

(II) $\vec{C X}$ .

(ii) Given that $Z$ is the point on $X C$ such that $\vec{X Z} = \frac{2}{3} \vec{X C}$ and that $\vec{Y C} = k \vec{A C}$ , express $\vec{Y Z}$ in terms of $k$ , $p$ and $q$ .

(iii) If, in addition, $YZ$ is parallel to $AX$ , find the numerical value of

(I) $k$ ,

(II) $\frac{A B}{Y Z}$ ,

(III) $\frac{Area of Δ}{Area of Δ} \frac{Y Z C}{A B C}$ .

10. $\vec{A B} = (\begin{matrix} 3 \\ −4 \end{matrix})$ , $\vec{A C} = (\begin{matrix} u \\ - & 12 \end{matrix})$ and $\vec{A D} = (\begin{matrix} - 8 \\ v \end{matrix})$ .

(a) Given that $C$ lies on $A B$ produced, write down the value of $u$ .

(b) Given that $AD$ is perpendicular to $AB$ , calculate the value of $v$ .

11. In the diagram, $\vec{O P} = p$ , $\vec{O Q} = q$ and $\vec{P X} = \vec{X Q}$ .

(a) Express in terms of $p$ and $q$

(i) $\vec{P X}$ ,

(ii) $\vec{O X}$ .

(b) Given that $\vec{O R} = \frac{3}{4} p + \frac{3}{4} q$ , write down the numerical value of the ratio $\frac{R X}{X O}$ .

(c) $\vec{Q R}$ is produced to point $Y$ where $\vec{Q Y} = k \vec{Q R}$ . Obtain expressions for $\vec{Q R}$ and $\vec{Q Y}$ and hence prove that $\vec{O Y} = \frac{3 h}{4} p + (1 - \frac{h}{4}) q$ .

(d) Given also that $\vec{O Y} = k \vec{O P}$ , form an equation involving $p$ , $q$ , $h$ and $k$ and use it to find the value of $h$ and $k$ .

(e) Deduce the numerical value of the ratio $\frac{Y P}{P O}$ .

12. The diagram in the answer space shows the origin $O$ and three points $A$ , $B$ and $P$ . The position vectors of $A$ and $B$ with respect to $O$ are $a$ and $b$ . Given that another point $Q$ is such that $\vec{O Q} = h a$ and $\vec{Q P} = k b$ ,

(a) mark on the diagram, and label clearly, the point $Q$ ,

(b) determine the value of

(i) $h$ ,

(ii) $k$ .

13. In the diagram, $P Q R S$ is a straight line and $PQ = QR = R S$ . Given that $O Q = 2 x + y$ and $P O = y - x$ , express, as simply as possible, in terms of $x$ and/or $y$ ,

(a) $RQ$ ,

(b) $OP$ ,

(c) $OS$ .

14. It is given that $A$ is the point $(- 2, 0)$ , $B$ is the point $(7, 0)$ and $\vec{B C} = (\begin{matrix} 3 \\ 2 \end{matrix})$ . Find

(a) $| \vec{A B} |$ ,

(b) the coordinates of the point $C$ ,

(c) the gradient of the line $A C$ .

15. (a) Given that $p = (\begin{matrix} 3 \\ 2 \end{matrix})$ and $q = (\begin{matrix} - 2 \\ 4 \end{matrix})$ ,

(i) express $3 p - \frac{1}{2} q$ as a column vector,

(ii) find $| q |$ .

(b) In the diagram, $PA = 5 a$ , $PB = 10 b$ and $X$ is a point on $A B$ such that $\frac{A X}{X B} = \frac{2}{3}$ . Express in terms of $a$ and $b$ , as simply as possible,

(i) $\vec{A B}$ ,

(ii) $\vec{A X}$ ,

(iii) $\vec{P X}$ .

The line $P X$ is produced to $C$ such that $\vec{B C} = 6 a - 2 b$ .

The lines $P A$ and $B C$ are produced to intersect at $Y$ .

(iv) Given that $\frac{B Y}{B C} = k$ , express $\vec{B Y}$ in terms of $k$ , $a$ and $b$ .

(v) Given that $\frac{P Y}{P A}$ $= r$ , express $\vec{P Y}$ in terms of $r$ , $a$ and $b$ .

(vi) Using the expressions for $\vec{P B}$ , $\vec{B Y}$ and $\vec{P Y}$ , form an equation connecting $k$ , $r$ , $a$ and $b$ . Use this equation to find the values of $k$ and $r$ .

16. $A B C D$ is a parallelogram, $M$ is the midpoint of $B C$ and $L$ is a point on $A D$ such that $3 A L = L D$ . Given that $\vec{A B} = p$ and $\vec{B M} = 2 q$ , express as simply as possible, in terms of $p$ and/or $q$ ,

(a) $\vec{M D}$ ,

(b) $\vec{D L}$ ,

(c) $\vec{L M}$ .

17. It is given that $A$ is the point $(3, 4)$ , $B$ is the point $(12, 10)$ and $P$ is the point on $A B$ such that $\vec{A P} = \frac{1}{2} \vec{P B}$ .

(a) Express as column vectors

(i) $\vec{A B}$ ,

(ii) $\vec{A P}$ ,

(iii) the position vector of $P$ relative to the origin $O$ .

(b) If $O$ , $A$ and $B$ are three of the vertices of a parallelogram, find the coordinates of two possible positions of the fourth vertex.

18. It is given that $\vec{O A} = (\begin{matrix} 3 \\ 4 \end{matrix})$ , $\vec{A B} = (\begin{matrix} 9 \\ n \end{matrix})$ and $\vec{O B} = (\begin{matrix} 2 m \\ - m \end{matrix})$ .

Find the value of

(a) $| \vec{O A} |$ ,

(b) $m$ ,

(c) $n$ .

19. In the triangle $A B C$ , $P$ is the mid-point of $B C$ . The line $R Q$ is drawn parallel to $B C$ intersecting $A P$ at $G$ . Given that $\vec{A G} = 2 p$ , $\vec{G P} = p$ and $\vec{G Q} = 2 q$ , express the following as simply as possible in terms of $p$ and/or $q$ .

(a) $\vec{P C}$ ,

(b) $\vec{A C}$ ,

(c) $\vec{P R}$ .

20. In the diagram, $M$ is the midpoint of $X Z$ , $\vec{O X} = p + 2 q$ , $\vec{O Z} = 7 p - 2 q$ and $\vec{Z Y} = 3 k q - p$ , where $k$ is a constant.

(a) Express as simply as possible in terms of $p$ and/or $q$ .

(i) $\vec{X Z}$ ,

(ii) $\vec{X M}$ ,

(iii) $\vec{O M}$ .

(b) Express $\vec{O Y}$ in terms of $p$ , $q$ and $k$ .

(c) If $Y$ lies on $O M$ produced, find the value of $k$ .

21. (a) The diagram shows three vectors $\vec{O A}$ , $\vec{O B}$ and $\vec{O C}$ . Given that $\vec{O T} = \vec{O A} + 2 \vec{O B} - 3 \vec{O C}$ , mark the point $T$ on the diagram, and label it clearly.

(b) The column vectors $u$ , $v$ and $w$ are defined by

$u = (\begin{matrix} 10 \\ 4 \end{matrix})$ , $v = (\begin{matrix} - 2 \\ 5 \end{matrix})$ , $w = (\begin{matrix} p \\ q \end{matrix})$ .

(i) Express $\frac{1}{2} u + 3 v$ as a column vector.

(ii) Given also that $u - w = w - v$ , find the value of $p$ and the value of $q$ .

22. $O A B C$ is a parallelogram and $M$ is the midpoint of $B C$ . $O C$ is produced to $D$ so that $\vec{O C} = \vec{C D}$ . $O M$ is produced to meet $D B$ at $N$ .

(a) Given that $\vec{O A} = 2 p$ and $\vec{A B} = 2 q$ , express the following vectors in terms of $p$ and $q$ , giving each of your answers in its simplest form.

(i) $\vec{O B}$ ,

(ii) $\vec{O M}$ ,

(iii) $\vec{B D}$ .

(b) Given that $\vec{O N} = h \vec{O M}$ and that $\vec{B N} = k \vec{B D}$ , use the fact that $\vec{O N} = \vec{O B} + \vec{B N}$ to write down an equation in terms of $h$ , $k$ , $p$ and $q$ . Hence show that $h = \frac{4}{3}$ and find the value of $k$ .

(c) Find the numerical value of

(i) $\frac{O M}{O N}$ ,

(ii) $\frac{the area of Δ}{the area of Δ} \frac{M D O}{M D N}$ ,

(iii) $\frac{the area of Δ}{the area of trapezium} \frac{O B D}{O A B D}$ .

23. $\vec{P Q} = (\begin{matrix} 2 \\ - 4 \end{matrix})$ , $\vec{Q R} = (\begin{matrix} - 8 \\ 6 \end{matrix})$ . Find

(a) $\vec{P R}$ ,

(b) $\vec{R X}$ , given that $2 \vec{R X} = \vec{Q R}$ ,

(c) $| \vec{Q R} |$ .

24. $A B C D$ is a parallelogram in which $\vec{A B} = p$ and $\vec{B C} = q$ . The point $E$ on $A D$ is such that $A E = \frac{1}{4} A D$ .

(a) Express, in terms of $p$ and/or $q$ ,

(i) $\vec{A C}$ ,

(ii) $\vec{A E}$ ,

(iii) $\vec{B E}$ .

(b) $A C$ and $B E$ intersect at $F$ . Given that $\vec{B F} = k \vec{B E}$ , express $\vec{B F}$ in terms of $p$ , $q$ and $k$ .

(c) Hence show that $\vec{A F} = (1 - k) p + \frac{1}{4} k q$ .

(d) Given also that $\vec{A F} = h \vec{A C}$ , express $\vec{A F}$ in terms of $p$ , $q$ and $h$ .

(e) Using these two expressions for $\vec{A F}$ , find the value of $h$ and the value of $k$ .

(f) Find the ratio $A F : F C$ .

25. The vectors $\vec{A B}$ and $\vec{C D}$ are such that $\vec{A B} = (\begin{matrix} - 6 \\ 8 \end{matrix})$ and $\vec{C D} = (\begin{matrix} u \\ 2 \end{matrix})$ .

(a) Find $| \vec{A B} |$ .

(b) Given that $\vec{C D}$ is parallel to $\vec{A B},$ find the value of $u$ .

26. In the diagram, $\vec{O P} = p$ , $\vec{O Q} = q$ and $R$ is the point on $P Q$ such that $P R = \frac{3}{4} P Q$ .

(a) Express as simply as possible in terms of $p$ and $q$ ,

(i) $\vec{P Q}$ ,

(ii) $\vec{P R}$ ,

(iii) $\vec{O R}$ .

(b) Given that $S$ is the point such that $\vec{P S} = 3 q$ and that $\vec{O S} = k \vec{O R}$ ,

(i) draw a sketch to show the approximate position of $S$ ,

(ii) find the value of $k$ ,

(iii) write down the numerical value of the ratio $O R : R S$ .

27. $P$ is the point $(3, 4)$ and $Q$ is the point $(11, 10)$ .

(a) Calculate the coordinates of the midpoint of $P Q$ .

(b) Express $\vec{P Q}$ as a column vector.

(c) Find the value of $| \vec{P Q} |$ .

28. (a) The point $O$ lies inside square $P Q R S$ . Given that $\vec{O P} = (\begin{matrix} - 4 \\ 2 \end{matrix})$ and $\vec{O Q} = (\begin{matrix} 3 \\ 5 \end{matrix})$ , express as column vectors

(i) $\vec{P Q}$ ,

(ii) $\vec{R S}$ ,

(iii) $\vec{Q R}$ .

(b) $O A X$ and $O B Z$ are straight lines and $A Z$ intersects $B X$ at $Y$ . $O B = B Z$ and $\frac{A Y}{Y Z} = \frac{1}{2}$ . Given that $\vec{O A} = 3 a$ and $\vec{O B} = 3 b$ , express as simply as possible in terms of $a$ and $b$ ,

(i) $\vec{A Z}$ ,

(ii) $\vec{A Y}$ ,

(iii) $\vec{O Y}$ ,

(iv) $\vec{B Y}$ .

Given that $\vec{B X} = h \vec{B Y}$ , write down an expression for $\vec{B X}$ and show that $\vec{O X} = 2 h a + (3 - h) b$ . Given also that $\vec{O X} = k \vec{O A}$ , form an equation involving $a$ , $b$ , $h$ and $k$ and use it to find $k$ . Hence write down the numerical value of the ratio $\frac{O A}{O X}$ .

29. $p = (\begin{matrix} 6 \\ - 8 \end{matrix})$ , $q = (\begin{matrix} 5 \\ 4 \end{matrix})$ , $r = (\begin{matrix} 9 \\ k \end{matrix})$ .

(a) Express $p + 2 q$ as a column vector.

(b) Find $| p |$ .

(c) Given that $r$ is parallel to $p$ , find the value of $k$ .

30. $A B C D$ is a parallelogram. The point $E$ , on $B D$ , is such that $B E = \frac{1}{4} B D$ .

(a) Given that $\vec{A B} = 4 p$ and $\vec{A D} = 4 q$ , express, in terms of $p$ and/or $q$ ,

(i) $\vec{C D}$ ,

(ii) $\vec{B D}$ .

(b) Show that $\vec{A E} = 3 p + q$ .

(c) The point $X$ lies at the point of intersection of $B C$ and $A E$ produced, so that $\vec{A X} = \frac{4}{3} \vec{A E}$ . Given that $\vec{B X} = h \vec{B C}$ , find the value of $h$ .

(d) Find the numerical value of $\frac{B X}{X C}$ .

31. It is given that $a = (\begin{matrix} 3 \\ - 4 \end{matrix})$ and $b = (\begin{matrix} - 1 \\ 1 \end{matrix})$ .

(a) Find $a + b$ .

(b) Calculate $| a |$ .

(c) If $a = 3 b + 2 c$ , express $c$ as a column vector.

32. $\vec{A B} = (\begin{matrix} 8 \\ - 6 \end{matrix})$ and $\vec{C D} = \frac{3}{2} \vec{A B}$ .

(a) Calculate $| \vec{A B} |$ .

(b) Express $\vec{C D}$ as a column vector.

(c) Given that $A$ is the point $(6, 9)$ find the coordinates of the point $B$ .

(d) Given that $D$ is the point $(1, 4)$ find the coordinates of the point $C$ .

33. In the diagram, $\vec{O A} = 3 x$ , $\vec{O B} = 2 x + y$ and $\vec{O C} = 2 y$ .

(a) Express $\vec{C O}$ in terms of $y$ .

(b) The point $D$ is such that $\vec{O D} = h \vec{O A}$ . Show that $\vec{C D} = 3 h x - 2 y$ .

(c) Express $\vec{C B}$ in terms of $x$ and $y$ , giving your answer in its simplest form.

(d) Given also that $\vec{C D} = k \vec{C B}$ , write down an equation in terms of $x$ , $y$ , $h$ and $k$ .

(e) Find the value of $h$ and the value of $k$ .

(f) Find $\vec{A D}$ .

34. The diagram shows the positions of the points $X$ , $P$ and $Q$ where $\vec{P Q} = (\begin{matrix} 4 \\ 2 \end{matrix})$ .

(a) $Y$ is the point such that $P Q Y X$ is a parallelogram. Express $\vec{Q Y}$ as a column vector.

(b) $Z$ is the point such that $P Q X Z$ is a parallelogram. Express $\vec{X Z}$ as a column vector.

(c) $R$ is the midpoint of $P Q$ . Express $\vec{X R}$ as a column vector.

35. $O P Q R$ is a parallelogram. $M$ is the midpoint of $P Q$ and $N$ is the midpoint of $Q R$ . $\vec{O P} = 2 a$ and $\vec{O R} = 2 b$ .

(a) Express in terms of $a$ and/or $b$

(i) $\vec{P M}$ ,

(ii) $\vec{O M}$ ,

(iii) $\vec{O N}$ ,

(iv) $\vec{P N}$ .

(b) $O M$ and $P N$ meet at $X$ . Given that $\vec{P X} = h \vec{P N}$ , express $\vec{P X}$ in terms of $a$ , $b$ and $h$ .

(c) Hence show that $\vec{O X} = (2 - h) a + 2 h b$ .

(d) Given also that $\vec{O X} = k \vec{O M}$ , express $\vec{O X}$ in terms of $a$ , $b$ and $k$ .

(e) Using these two expressions for $\vec{O X}$ , find the value of $h$ and the value of $k$ .

(f) Find the ratio $X N : P N$ .

36. $X$ lies on the straight line $A B$ and $A X : X B = 1 : 3$ . With respect to an origin $O$ , the position vector of $A$ is $4 p$ and the position vector of $B$ is $4 q$ . Express in terms of $p$ and $q$ ,

(a) $\vec{A B}$ ,

(b) $\vec{A X}$ ,

(c) $\vec{O X}$ , giving this last answer in its simplest form.

37. (a) It is given that $p = (\begin{matrix} 6 \\ 5 \end{matrix})$ , $q = (\begin{matrix} - 4 \\ 3 \end{matrix})$ and $r = (\begin{matrix} a \\ b \end{matrix})$ .

(i) Find $| q |$ .

(ii) Express $2 p + 3 q$ as a column vector.

(iii) Given that $p - q = 2 r$ , find the value of $a$ and the value of $b$ .

(b) $A B C D$ is a quadrilateral in which $\vec{A B} = 2 s$ , $\vec{D C} = 6 s$ and $\vec{D A} = 2 t$ .

(i) Sketch the quadrilateral $A B C D$ .

(ii) Consider the pair of sides $A B$ and $D C$ . Write down two important facts about this pair of sides.

(iii) Express $\vec{B C}$ , as simply as possible, in terms of $s$ and/or $t$ .

(iv) The sides $D A$ and $C B$ , when produced, meets at $X$ . Express $\vec{X A}$ , as simply as possible, in terms of $s$ and/or $t$ .

38. $O A B C$ is a parallelogram. The point $X$ on $A C$ is such that $A X = \frac{1}{5} A C$ . The point $Y$ on $A B$ is such that $A Y = \frac{1}{4} A B$ . Given that $\vec{O A} = 20 p$ and $\vec{O C} = 20 q$ , express in terms of $p$ and $q$

(a) $\vec{A C}$ ,

(b) $\vec{A X}$ ,

(c) $\vec{O X}$ ,

(d) $\vec{O Y}$ .

What do the results of (c) and (d) tell you about $O$ , $X$ and $Y$ ?

39. In the diagram, $A C$ is parallel to $O D$ , $A$ is the midpoint of $O B$ and $X$ is the point on $A D$ such that $A X = \frac{1}{3} A D$ . $\vec{O A} = (\begin{matrix} 5 \\ 0 \end{matrix})$ and $\vec{O D} = (\begin{matrix} 8 \\ 6 \end{matrix})$ .

(a) Find the value of $| \vec{O D} |$ .

(b) Express each of the following as a column vector:

(i) $\vec{C A}$ ,

(ii) $\vec{A D}$ ,

(iii) $\vec{B X}$ .

40. The diagram represents an equilateral triangle $A B C$ with sides of length 12 cm. $M$ is the midpoint of $A C$ , and $C T = 4 cm$ . It is given that $\vec{C B} = 3 p$ and $\vec{M C} = q$ .

(a) Express, as simply as possible, in terms of $p$ and/or $q$ ,

(i) $\vec{A C}$ ,

(ii) $\vec{C T}$ ,

(iii) $\vec{A T}$ ,

(iv) $\vec{M B}$ .

(b) $X$ is the point where $B M$ intersects $A T$ .

(i) Given that $\vec{M X} = h \vec{M B}$ , show that $\vec{A X} = 3 h p + (h + 1) q$ .

(ii) Given also that $\vec{A X} = k \vec{A T}$ , form an equation involving $p$ , $q$ , $h$ and $k$ .

(iii) Use this equation to find the value of $h$ .

(iv) By using the Pythagoras' Theorem in $Δ A B M$ , calculate $B M$ . Hence write down the length of $X M$ , giving your answer in centimetres correct to one decimal place.

41. $\vec{A B} = (\begin{matrix} - 8 \\ 6 \end{matrix})$ .

(a) Calculate $| \vec{A B} |$ .

(b) Given that $A$ is the point $(7, 3)$ , find the coordinates of the point $B$ .

(c) It is given that $\vec{C D}$ is parallel to $\vec{A B}$ and that it is half as long as $\vec{A B}$ . Express $\vec{C D}$ as a column vector.

42. In the diagram, $X V Z$ is a straight line, $\vec{X Y} = 8 p$ , $\vec{X Z} = 4 p + 9 q$ and $\vec{Y V} = - 6 p + c q$ .

(a) Express $\vec{V Z}$ in terms of $p$ , $q$ and $c$ .

(b) Given that $\vec{Y V} = h \vec{Y Z}$ , form an equation involving $p$ , $q$ , $h$ and $c$ , and use it to find the numerical values of $h$ and $c$ .

(c) The point $K$ is outside triangle $X Y Z$ and is such that $\vec{X K} = - 4 p + 3 q$ . Is $\vec{X K}$ parallel to $\vec{Y V}$ ? Justify your answer.

43. $\vec{P O} = (\begin{matrix} - 4 \\ 7 \end{matrix})$ , $\vec{Q R} = (\begin{matrix} 1 \\ 2 \end{matrix})$ , $\vec{R S} = (\begin{matrix} h \\ 10.5 \end{matrix})$ .

(a) Express as a column vector $\vec{P Q} + 3 \vec{Q R}$ .

(b) Given that $\vec{R S}$ is parallel to $\vec{P Q}$ , find the value of $h$ .

(c) Find $| \vec{P Q} |$ , giving your answer correct to the nearest whole number.

44. In the diagram, the lines $S T$ , $P Q$ and $O R$ are parallel. $O P S$ , $O Q T$ and $R Q S$ are straight lines.

(a) Given that $\vec{O R} = 3 a$ , $\vec{O P} = b$ , $\vec{P S} = 2 b$ and $\vec{O Q} = 2 a + b$ express, in terms of $a$ and/or $b$ ,

(i) $\vec{P Q}$ ,

(ii) $\vec{Q R}$ ,

(iii) $\vec{Q T}$ ,

(iv) $\vec{S T}$ .

(b) Find the numerical value of

(i) $\frac{the area of Δ O P Q}{the area of Δ S P Q}$ ,

(ii) $\frac{the area of Δ O P Q}{the area of Δ Q S T}$ ,

(iii) $\frac{the area of Δ O P Q}{the area of Δ O R Q}$ .

45. In the diagram, $\vec{A B} = p$ , $\vec{A C} = q$ and $X$ is the point on $C B$ such that $C X = \frac{1}{3} C B$ .

(a) Express as simply as possible in terms of $p$ and/or $q$ ,

(i) $\vec{C B}$ ,

(ii) $\vec{A X}$ .

(b) Given further that $\vec{A X} = h p + k q$ , and that $T$ is the point such that $\vec{A T} = h p$ , mark and label the point $T$ on the diagram in the answer space.

46. In the diagram, $\vec{O P} = a$ , $\vec{O Q} = b$ and $X$ is the point on $P Q$ such that $\vec{P X} = \frac{1}{4} \vec{P Q}$ .

(a) Express as simply as possible in terms of $a$ and/or $b$ ,

(i) $\vec{P Q}$ ,

(ii) $\vec{O X}$ .

(b) Given further that $\vec{O X} = h a + k b$ , and that $N$ is the point such that $\vec{O N} = h a$ , mark and label the point $N$ on the diagram in the answer space.

47. $\vec{A B} = (\begin{matrix} - 6 \\ 8 \end{matrix})$ .

(a) Calculate $| \vec{A B} |$ .

(b) Given that $A B C D$ is a parallelogram, express $\vec{C D}$ as a column vector.

(c) Given that $A$ is the point $(7, 3)$ and that $M$ is the midpoint of $A B$ , find the coordinates of $M$ .

48. In the diagram $O A = \frac{1}{3} O P$ , and $O B = \frac{1}{4} O Q$ . $M$ is the midpoint of $O Q$ , and $\vec{M X} = \frac{1}{5} \vec{M P}$ .

(a) Given that $\vec{O A} = a$ and $\vec{O B} = b$ express, as simply as possible, in terms of $a$ and/or $b$ ,

(i) $\vec{O P}$ ,

(ii) $\vec{O M}$ ,

(iii) $\vec{A Q}$ ,

(iv) $\vec{M P}$ ,

(v) $\vec{M X}$ ,

(vi) $\vec{A X}$ .

(b) Prove that $A X$ , when produced, will pass through $Q$ .

(c) Find the ratio $A X : X Q$ .

(d) Given that the area of $Δ O P Q$ is 30 cm², calculate the area of

(i) $Δ P M Q$ ,

(ii) $Δ P Q X$ ,

(iii) $Δ O A B$ .

49. It is given that $\vec{O P} = p$ and $\vec{O Q} = q$ . $R$ , $S$ , $T$ and $U$ are points on the grid. Express, in terms of $p$ and/or $q$ ,

(a) $\vec{O R}$ ,

(b) $\vec{O S}$ ,

(c) $\vec{O T}$ ,

(d) $\vec{O U}$ .

50. In triangle $O R S$ , the point $A$ on $O R$ is such that $O A = 2 A R$ . $B$ is the midpoint of $O S$ , $X$ is the midpoint of $A B$ and $O X$ produced meets $R S$ at $Y$ . $\vec{O A} = 2 p$ and $\vec{O B} = 2 q$ .

(a) Express in terms of $p$ and/or $q$

(i) $\vec{A B}$ ,

(ii) $\vec{A X}$ ,

(iii) $\vec{O X}$ ,

(iv) $\vec{R S}$ .

(b) Given that $\vec{R Y} = h \vec{R S}$ , express $\vec{R Y}$ in terms of $p$ , $q$ and $h$ .

(c) Hence show that $\vec{O Y} = 3 (1 - h) p + 4 h q$ .

(d) Given also that $\vec{O Y} = k \vec{O X}$ , express $\vec{O Y}$ in terms of $p$ , $q$ and $k$ .

(e) Using these two expressions for $\vec{O Y}$ , find the value of $h$ and the value of $k$ .

(f) Find the ratio $R Y : Y S$ .

(g) Express $\vec{X Y}$ in terms of $p$ and $q$ .

51. $\vec{A B} = (\begin{matrix} 7 \\ 1 \end{matrix})$ , $\vec{B C} = (\begin{matrix} 2 \\ - 5 \end{matrix})$ .

(a) Express $\vec{A C}$ as a column vector.

(b) $D$ is the point $(- 2, 1)$ and $E$ is $(h, 6)$ .

(i) Express $\vec{D E}$ as a column vector.

(ii) If $\vec{D E}$ is parallel to $\vec{A B}$ , find the value of $h$ .

(iii) If instead, $| \vec{D E} | = | \vec{A B} |$ , find the two possible values of $h$ .

52. In the diagram, $A B C D$ is a parallelogram, $X$ is the midpoint of $A C$ , $Y$ is the midpoint of $A X$ and $W$ is the point on $A D$ such that $A W = 2 W D$ .

(a) Given that $\vec{A B} = p$ and $\vec{A D} = 3 q$ express, as simply as possible, in terms of $p$ and/or $q$ ,

(i) $\vec{A C}$ ,

(ii) $\vec{A Y}$ ,

(iii) $\vec{B Y}$ ,

(iv) $\vec{X W}$ .

(b) What do your answers to (a)(iii) and (a)(iv) tell you about $\vec{B Y}$ and $\vec{X W}$ ?

(c) What is the special name given to the quadrilateral $B X W Y$ ?

(d) Write down the value of each of the following.

(i) $\frac{Δ A B Y}{Δ B C Y}$ ,

(ii) $\frac{Δ B C Y}{Δ A X W}$ .

53. $\vec{O A} = (\begin{matrix} - 2 \\ 1 \end{matrix})$ and $\vec{O B} = (\begin{matrix} 3 \\ 4 \end{matrix})$ .

(a) Given that $\vec{O C} = \vec{O A} + \vec{O B}$ , express $\vec{O C}$ as a column vector.

(b) Express $\vec{A B}$ as a column vector.

(c) If $\vec{L M} = (\begin{matrix} 4 \\ - 2 \end{matrix})$ , what is the special name given to the quadrilateral $A L M O$ ?

54. In the diagram $\vec{O A} = 2 p$ , $\vec{O B} = 3 q$ and $\vec{B X} = p - q$ . The lines $O X$ and $A B$ intersect at $L$ .

(a) Express as simply as possible in terms of $p$ and/or $q$

(i) $\vec{O X}$ ,

(ii) $\vec{A B}$ .

(b) Given that $\vec{A L} = h \vec{A B}$ , express $\vec{A L}$ in terms of $p$ , $q$ and $h$ .

(c) Hence show that $\vec{O L} = (2 - 2 h) p + 3 h q$ .

(d) Given also that $\vec{O L} = k \vec{O X}$ , form an equation involving $p$ , $q$ , $h$ and $k$ and use it to find the numerical value of $h$ .

55. Given that $\vec{O P} = u$ , $\vec{O Q} = v$ and $\vec{P R} = w$ , express $\vec{Q R}$ in terms of $u$ , $v$ and $w$ .

56. Given that $P$ is the point $(1, 1)$ , $\vec{P Q} = (\begin{matrix} - 3 \\ 2 \end{matrix})$ , $\vec{P R} = (\begin{matrix} 5 \\ 4 \end{matrix})$ and that $T$ is the midpoint of $Q R$ , find

(a) $\vec{Q R}$ ,

(b) $\vec{P T}$ ,

(c) the coordinates of the point $X$ such that $P Q X R$ is a parallelogram.

57. The grid in the working space is provided for you to use if you wish. $P$ is the point $(1, 1)$ and $Q$ is the point $(3, 2)$ .

(a) Write down $\vec{P Q}$ as a column vector.

(b) The line $P Q$ is mapped onto the line $P_{1} Q_{1}$ by an enlargement of scale factor 3.

(i) Write down $\vec{P_{1} Q_{1}}$ as a column vector.

(ii) Given that $P_{1}$ is the point $(1, 5)$ , find the coordinates of the centre of the enlargement.

58. In the diagram $Q R = 4 Q S$ and $S P = 5 S X$ . $T$ is the midpoint of $P R$ . $\vec{Q S} = a$ and $\vec{R T} = b$ .

(a) Express, as simply as possible, in terms of $a$ and/or $b$ ,

(I) $\vec{S R}$ ,

(II) $\vec{S P}$ ,

(III) $\vec{S X}$ .

(b) Show that $\vec{Q X} = \frac{2}{5} (4 a + b)$ .

(c) Express $\vec{Q T}$ as simply as possible, in terms of $a$ and $b$ .

(d) Calculate the value of

(i) $\frac{Q X}{Q T}$ ,

(ii) $\frac{Δ P Q X}{Δ P Q T}$ ,

(iii) $\frac{Δ P Q X}{Δ P Q R}$ .

59. The grid below is provided for you to use in this question, if you wish.

$C$ is the point $(1, 2)$ and $D$ is the point $(4, 3)$ .

(a) Write down $\vec{C D}$ as a column vector.

(b) The line $C D$ is mapped onto the line $C_{1} D_{1}$ by an enlargement of scale factor 3.

(i) Write down $\vec{C_{1} D_{1}}$ as a column vector.

(ii) Given that $C_{1}$ is the point $(1, 6)$ , find the coordinates of the centre of the enlargement.

60. In the diagram, $A$ is $(0, 4)$ , $B$ is $(3, 6)$ and $O$ is the origin.

(a) Express $\vec{A B}$ as a column vector.

(b) Calculate the coordinates of the point $D$ , where $\vec{B D} = 2 \vec{A B}$ .

(c) $\vec{A C} = (\begin{matrix} 5 \\ - 2 \end{matrix})$ .

(i) Calculate the length of $A C$ .

(ii) Write down the gradient of the line $A C$ .

(iii) Write down the equation of the line $A C$ .

(iv) The point $E$ lies on $A C$ and $E B$ is parallel to the $y$ -axis. Calculate the coordinates of $E$ .

61. The quadrilateral $A B C D$ is such that $\vec{A B} = 2 p$ , $\vec{B C} = q$ and $\vec{C D} = - 3 p$ .

(a) What is the special name given to the quadrilateral $A B C D$ ?

(b) Express $\vec{D A}$ in terms of $p$ and $q$ , giving your answer in its simplest form.

62. $O A C B$ is a quadrilateral. $O B$ is parallel to $A C$ . $D$ is the point on $B C$ such that $B D = \frac{1}{3} B C$ . The lines $O B$ and $A D$ produced meet at $F$ .

(a) Given also that $\vec{A C} = (\begin{matrix} - 8 \\ 6 \end{matrix})$ , calculate the value of $| \vec{B F} |$ .

(b) Given also that $O B = A C$ , and $\vec{O A} = (\begin{matrix} 5 \\ 2 \end{matrix})$ , express $\vec{F C}$ as a column vector.

63. In the diagram, $O A B C D E$ is a regular hexagon. $\vec{O A} = a$ and $\vec{O C} = c$ . Express the following vectors, as simply as possible, in terms of $a$ and/or $c$ .

(i) $\vec{C D}$ ,

(ii) $\vec{A C}$ ,

(iii) $\vec{O B}$ .

64. On the grid in the answer space $\vec{O A} = a$ and $\vec{O B} = b$ . The point $Y$ is also marked.

(a) On the grid mark clearly the point $X$ , such that $\vec{O X} = 3 b - 2 a$ .

(b) Write down $\vec{O Y}$ in terms of $a$ and $b$ .

(c) It is given that $\vec{O P} = a + n b$ , where $n$ takes all values from $- 3$ to $3$ . On the grid draw the locus of $P$ .

65. On the grid in the answer space $\vec{O P} = p$ and $\vec{O Q} = q$ . The point $R$ is also marked.

(a) Write down $\vec{O R}$ in terms of $p$ and $q$ .

(b) On the grid mark clearly the point $S$ , such that $\vec{O S} = - 2 p - 3 q$ .

(c) It is given that $\vec{O T} = n p + 2 q$ , where $n$ takes all values from $- 3$ to $4$ . On the grid draw the locus of $T$ .

66. On the grid in the answer space, $\vec{O P} = p$ and $\vec{O Q} = q$ .

(a) Mark clearly on the grid the point $X$ such that $\vec{O X} = 2 p$ and the point $Y$ such that $\vec{O Y} = 2 p - 3 q$ .

(b) What is the special name given to the quadrilateral $O Q X Y$ ?

67. (a) Given that $\vec{A B} = (\begin{matrix} 2 \\ 3 \end{matrix})$ , $\vec{B C} = (\begin{matrix} 0 \\ 1 \end{matrix})$ and $\vec{C D} = (\begin{matrix} - 5 \\ 1 \end{matrix})$ , find $\vec{D A}$ .

68. In the diagram, $O A B C$ is a parallelogram. The position vectors of the points $A$ and $B$ are given by $\vec{O A} = (\begin{matrix} 5 \\ 1 \end{matrix})$ , $\vec{O B} = (\begin{matrix} 8 \\ 6 \end{matrix})$ .

(a) Find $| \vec{O B} |$ .

(b) Express as column vectors

(i) $\vec{A B}$ ,

(ii) $\vec{C A}$ .

(c) The point $P$ lies on $C A$ produced and $\vec{A P} = h \vec{C A}$ .

(i) Show that $\vec{O P} = (\begin{matrix} 5 + 2 h \\ 1 - 4 h \end{matrix})$ .

(ii) Given that $P$ lies on the $x$ -axis, find

(a) the value of $h$ ,

(b) the coordinates of $P$ .

69. In the diagram, $O P Q R$ is a rhombus. The position vectors of the points $P$ and $R$ are given by $\vec{O P} = (\begin{matrix} 2 \\ 5 \end{matrix})$ and $\vec{O R} = (\begin{matrix} 5 \\ 2 \end{matrix})$ .

(a) Express as column vectors

(i) $\vec{P Q}$ ,

(ii) $\vec{O Q}$ ,

(iii) $\vec{P R}$ .

(b) The point $S$ lies on $P R$ produced and $\vec{R S} = k \vec{P R}$ .

(i) Show that $\vec{O S} = (\begin{matrix} 5 + 3 k \\ 2 - 3 k \end{matrix})$ .

(ii) Given that $S$ lies on the $x$ -axis, find

(a) the value of $k$ ,

(b) the coordinates of $S$ .

70. Given that $u = (\begin{matrix} 6 \\ - 8 \end{matrix})$ , $v = (\begin{matrix} - 9 \\ 10 \end{matrix})$ and $w = (\begin{matrix} 15 \\ p \end{matrix})$ , find

(a) (i) $| u |$ ,

(ii) $2 u + v$ .

(b) Given that the vector $w$ is parallel to the vector $u$ , calculate the value of $p$ .

71. In the diagram, $\vec{O A} = (\begin{matrix} 8 \\ 6 \end{matrix})$ and $\vec{O B} = (\begin{matrix} 3 \\ - 2 \end{matrix})$ .

(a) Find

(i) $| \vec{O A} |$ ,

(ii) $\vec{B A}$ .

(b) Given that $\vec{B C} = (\begin{matrix} x \\ 9 \end{matrix})$ and $\vec{B C} = k \vec{O A}$ ,

find the value of

(i) $k$ ,

(ii) $x$ ,

(iii) the ratio $\frac{area of triangle OAB}{area of triangle ACB}$ .

72. In the diagram, $\vec{O P} = (\begin{matrix} - 4 \\ 3 \end{matrix})$ and $\vec{O Q} = (\begin{matrix} 8 \\ 6 \end{matrix})$ .

(a) Find

(i) $| \vec{O P} |$ ,

(ii) $\vec{P Q}$ ,

(iii) $\vec{Q P}$ .

(b) Given that $\vec{P R} = (\begin{matrix} 6 \\ y \end{matrix})$ and $\vec{P R} = k \vec{O Q}$ , find the value of

(i) $k$ ,

(ii) $y$ .

73. $A$ is the point $(7, 3)$ and $B$ is the point $(5, 11)$ . Find

(a) the coordinates of the midpoint of $A B$ ,

(b) the vector $\vec{A B}$ .

74. (a) $p = (\begin{matrix} 4 \\ - 3 \end{matrix})$ , $q = (\begin{matrix} - 2 \\ 9 \end{matrix})$ and $r = (\begin{matrix} 1 \\ - 2 \end{matrix})$ .

(i) Find $| q |$ .

(ii) Express as a column vector

(a) $2 p + q$ ,

(b) $p - 2 r$ .

(iii) Write down two facts about the vectors $2 p + q$ and $p - 2 r$ .

(b)

In the diagram, $\vec{O A} = a$ and $\vec{O B} = b$ . $\vec{O M} = 3 \vec{O A}$ and $\vec{O N} = 2 \vec{O B}$ . $C$ is the point on $M N$ produced where $M N = N C$ .

(i) Express, as simply as possible, in terms of $a$ and/or $b$ ,

(a) $\vec{M O}$ ,

(b) $\vec{M N}$ ,

(c) $\vec{A B}$ ,

(d) $\vec{A C}$ .

(ii) Write down two facts which your answers to (c) and (d) tell you about $A$ , $B$ and $C$ .

75. $p = (\begin{matrix} 1 \\ - 4 \end{matrix})$ , $q = (\begin{matrix} - 3 \\ 4 \end{matrix})$ and $r = (\begin{matrix} m \\ 2 \end{matrix})$ .

(a) Find $| q |$ .

(b) Express $2 p - q$ as a column vector.

(c) Given that $p$ is parallel to $r$ , find $m$ .

76. $p = (\begin{matrix} 4 \\ - 3 \end{matrix})$ , $q = (\begin{matrix} - 2 \\ 1 \end{matrix})$ and $r = (\begin{matrix} k \\ 3 \end{matrix})$ .

(a) Find $| p |$ .

(b) Express $3 q - p$ as a column vector.

(c) Given that $q$ is parallel to $r$ , find $k$ .

77. In the diagram, $\vec{A D} = p$ , $\vec{D B} = q$ and $A C = 2 A D$ .

(a) Express, as simply as possible, in terms of $p$ and $q$ ,

(i) $\vec{A B}$ ,

(ii) $\vec{C B}$ .

(b) $E$ is a point such that $\vec{B E} = p$ .

(i) Express $\vec{A E}$ in terms of $p$ and $q$ .

(ii) Explain why $A B E C$ is a trapezium.

(c) Given that $| p | = 8$ , $| q | = 7$ and $A \hat{D} B = 57^{\circ}$ , calculate

(i) the perpendicular distance from $B$ to $A D$ ,

(ii) the area of the trapezium $A B E C$ ,

(iii) the length of $A B$ .

78. In the diagram, $O A B$ is a triangle.

$C$ is the point on $A B$ such that $A C : C B = 2 : 1$ .
The side $O B$ is produced to the point $D$ such that $\vec{O B} : \vec{B D} = 3 : 2$ .

It is given that $\vec{O A} = a$ and $\vec{O B} = b$ .

(a) Express, as simply as possible, in terms of $a$ and/or $b$ ,

(i) $\vec{A B}$ ,

(ii) $\vec{A C}$ ,

(iii) $\vec{O C}$ ,

(iv) $\vec{O D}$ .

(b) Show that $\vec{C D} = b - \frac{1}{3} a$ .

(c) It is given that $E$ is the point on $O A$ such that $\vec{O E} = \frac{5}{9} a$ .

Express, as simply as possible, in terms of $a$ and $b$ , the vector $\vec{E D}$ .

(d) (i) Show that $\vec{E D} = k \vec{C D}$ , where $k$ is a constant.

(ii) Write down two facts about $E D$ and $C D$ .

(e) Calculate $\frac{the area of Δ AEC}{the area of Δ OEC}$ .

Unit 7

Vectors