Vectors (Cambridge International IGCSE Mathematics): Exam Questions

$\vec{VW} = ( \begin{gathered} 10 \\ - 24 \end{gathered} )$ Find $\left| \vec{VW} \right| .$

$\vec{AB} = ( 6

• 1 )$$\vec{BC} = ( - 2 5 )$$\vec{DC} = ( 2
• 3 )$

Find

i) $\overset{\rightarrow}{AC}$,

$\vec{AC} = (

)$ [2]

ii) $\overset{\rightarrow}{BD}$,

$\vec{BD} = (

)$ [2]

iii) $\left| \vec{BC} \right|$.

$p = ( \begin{gathered} 4 \\ 5 \end{gathered} )$ $q = ( \begin{gathered} - 2 \\ 7 \end{gathered} )$

i) Find $2p+q.$

$( \begin{gathered} \\ \end{gathered} )$ [2]

ii) Find $\left| p \right|$.

$a = ( \begin{gathered} - 3 \\ 2 \end{gathered} ) b = ( \begin{gathered} 5 \\ 4 \end{gathered} ) c = ( \begin{gathered} 14 \\ 9 \end{gathered} )$

i) Find $3a-2b$ .

$( \begin{gathered} \\ \end{gathered} )$ [2]

ii) Find $\vert a \vert$.

i) Find the column vector $\overset{\rightarrow}{AB}$.

$\vec{AB} = ( \begin{gathered} \\ \end{gathered} )$ [1]

ii) Find $\vert \vec{AB} \vert$.

$\vert \vec{AB} \vert =$................................................. [2]

$\vec{OA} = ( \begin{gathered} 4 \\ 3 \end{gathered} )$ $\vec{AB} = ( \begin{gathered} 8 \\ - 7 \end{gathered} )$ $\vec{AC} = ( \begin{gathered} - 3 \\ 6 \end{gathered} )$

Find

i) $\left| \vec{OB} \right|$,

$\left| \vec{OB} \right| =$............................................... [3]

ii) $\overset{\rightarrow}{BC}$.

$\overset{\rightarrow}{BC}=\quad ( \begin{gathered} \\ \end{gathered} )$ [2]

Find the magnitude of the vector $( \begin{gathered} - 1 \\ 7 \end{gathered} )$.

The diagram shows triangle $OPT$.

In the diagram $\overset{\rightarrow}{OT}=t$ and $\overset{\rightarrow}{OP}=p$. $OK:KT=2:1$ and $TL:LP=2:1$.

Find, in terms of t and p, in its simplest form

i) $\overset{\rightarrow}{PL}$

ii) $\overset{\rightarrow}{KL}$

Extract:

The diagram shows triangle ![O P T].

In the diagram ![stack O T with rightwards arrow on top space equals space bold t] and ![stack O P with rightwards arrow on top space equals space bold p]. ![O K space colon space K T space equals space 2 space colon 1] and ![T L space colon space L P space equals space 2 space colon 1].

$KL$ is extended to the point $M$.

$\overset{\rightarrow}{KM}=-\frac{2}{3}t+\frac{4}{3}p.$

Show that $M$ lies on $OP$ extended.

P is the point (16, 9) and Q is the point (22, 24).

N is the point on PQ such that PN = 2NQ. Find the co-ordinates of N.

(.................... , ....................)

In the diagram, $O$ is the origin, $OT=2TD$ and $M$ is the midpoint of $TC$. $\overset{\rightarrow}{OC}=c$ and $\overset{\rightarrow}{OD}=d$.

Find the position vector of $M$. Give your answer in terms of $c$ and $d$ in its simplest form.

$a = ( \begin{gathered} - 3 \\ 2 \end{gathered} ) b = ( \begin{gathered} 5 \\ 4 \end{gathered} ) c = ( \begin{gathered} 14 \\ 9 \end{gathered} )$

$ma+nb=\text{c}$

Write down two simultaneous equations and solve them to find the value of $m$ and the value of $n$. Show all your working.

$m$ = ................................................
$n$ = ................................................

$OPTR$is a trapezium.

$\overset{\rightarrow}{OP}=a\,\overset{\rightarrow}{PT}=b\,\overset{\rightarrow}{OR}=3b$

i) Find $\overset{\rightarrow}{OT}$ in terms of $a$ and$b$

ii) Find $\overset{\rightarrow}{PR}$ in terms of $a$ and $b$ Give your answer in its simplest form.

Extract:

OPTR is a trapezium.

$\vec{OP} = \mathbf{a}$ $\vec{PT} = \mathbf{b}$ $\vec{OR} = 3\mathbf{b}$

ii) Find $\vec{PR}$ in terms of $\mathbf{a}$ and $\mathbf{b}$. Give your answer in its simplest form.

$S$ is the point on $PR$ such that $PS:SR$ = 1 : 3

Find $\overset{\rightarrow}{OS}$ in terms of $a$ and $b$. Give your answer in its simplest form.

Extract:

![S] is the point on ![P R] such that ![P S space colon space S R] = 1 : 3

Find ![stack O S with rightwards arrow on top] in terms of ![bold a] and ![bold b].

Give your answer in its simplest form.

What does your answer to part (b) tell you about the position of point $S$?

In the diagram, $OAB$ and $OED$ are straight lines. $O$ is the origin, $A$ is the midpoint of $OB$ and $E$ is the midpoint of $AC$. $\overset{\rightarrow}{AC}=a$and $\overset{\rightarrow}{CB}=b$.
Find, in terms of $a$ and $b$, in its simplest form

i) $\overset{\rightarrow}{AB}$,

$\overset{\rightarrow}{AB}=$................................................ [1]

ii) $\overset{\rightarrow}{OE}$,

$\overset{\rightarrow}{OE}=$................................................ [2]

iii) the position vector of $D$.

$OPQR$ is a rectangle and $O$ is the origin. $M$ is the midpoint of $RQ$ and $PT:TQ=2:1$. $\overset{\rightarrow}{OP}=p$ and $\overset{\rightarrow}{OR}=r$.

Find, in terms of $p$ and/or $r$, in its simplest form

i) $\overset{\rightarrow}{MQ}$,

$\overset{\rightarrow}{MQ}=$ .............................................. [1]

ii) $\overset{\rightarrow}{MT}$,

$\overset{\rightarrow}{MT}=$ .............................................. [1]

iii) $\overset{\rightarrow}{OT}$,

$\overset{\rightarrow}{OT}=$ .............................................. [1]

Extract:

[O P Q R] is a rectangle and [O] is the origin.

[M] is the midpoint of [R Q] and [P T : T Q = 2 : 1].

[O P = bold p] and [O R = bold r].

Find, in terms of [bold p] and/or [bold r], in its simplest form

i) [M Q =] .............................................. [1]

ii) [M T =] .............................................. [1]

iii) [O T =] .............................................. [1]

$R Q$ and $O T$ are extended and meet at $U$.

Find the position vector of $U$ in terms of $p$ and $r$. Give your answer in its simplest form.

$OAB$ is a triangle and $C$ is the mid-point of $OB$. $D$ is on $AB$ such that $AD$ : $DB$ = 3 : 5. $OAE$ is a straight line such that $OA:AE$ = 2 : 3. $\overset{\rightarrow}{OA}=a$ and $\overset{\rightarrow}{OC}=c$.

Find the following vectors, in terms of $a$ and $c$, in their simplest form

i) $\overset{\rightarrow}{AB}$,

$\overset{\rightarrow}{AB}=$................................................ [1]

ii) $\overset{\rightarrow}{AD}$,

$\overset{\rightarrow}{AD}=$................................................ [1]

iii) $\overset{\rightarrow}{CE}$,

$\overset{\rightarrow}{CE}=$................................................ [1]

iv) $\overset{\rightarrow}{CD}$.

$\overset{\rightarrow}{CD}=$................................................ [2]

Extract:

Find the following vectors, in terms of bold a and bold c, in their simplest form

i) stack A B with rightwards arrow on top,

stack A B with rightwards arrow on top equals................................................ [1]

ii) stack A D with rightwards arrow on top,

stack A D with rightwards arrow on top equals................................................ [1]

iii) stack C E with rightwards arrow on top,

stack C E with rightwards arrow on top equals................................................ [1]

iv) stack C D with rightwards arrow on top.

stack C D with rightwards arrow on top equals................................................ [2]

$\overset{\rightarrow}{CE}=k\overset{\rightarrow}{CD}$

Find the value of $k$.

$k$ = ................................................

$ABC$is a straight line.

$AB:BC=2:5$

$\overset{\rightarrow}{OA}=2a+b\,\overset{\rightarrow}{OB}=3a+2b$

Express $\overset{\rightarrow}{OC}$ in terms of $a$and $b$. Give your answer in its simplest form.

The diagram shows triangle $OAB$.

In the diagram $\overset{\rightarrow}{OA}=a$ and $\overset{\rightarrow}{OB}=b$. $OM:MA=3:1$, and $N$ is the midpoint of $AB$.

Find, in terms of $a$ and $b$, in its simplest form

(i) $\overset{\rightarrow}{NB}$

(ii) $\overset{\rightarrow}{MN}$

Extract:

The diagram shows triangle $O A B$ .

In the diagram $\overset{\rightarrow}{O A} = \mathbf{a}$ and $\overset{\rightarrow}{O B} = \mathbf{b}$ .

$O M : M A = 3 : 1$ , and $N$ is the midpoint of $A B$.

$MN$ is extended to the point $P$. $\overset{\rightarrow}{MP}=-\frac{3}{4}a+\frac{3}{2}b$.

Show that $P$ lies on $OB$ extended.