Vectors (Cambridge (CIE) O Level Maths): Exam Questions

The diagram shows triangle $OPT$.

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

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

i)   $\overrightarrow{PL}$

[2]

ii)  $\overrightarrow{KL}$

[2]

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

$\overrightarrow{KM}=-\frac{2}{3}\mathbf{t}+\frac{4}{3}\mathbf{p}.$

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 $TC$.
$\overrightarrow{OC}=\mathbf{c}$ and $\overrightarrow{OD}=\mathbf{d}$.  

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

$\mathbf{a}=\left(\begin{array}{c}-3\\ 2\end{array}\right)\mathbf{b}=\left(\begin{array}{c}5\\ 4\end{array}\right)\mathbf{c}=\left(\begin{array}{c}14\\ 9\end{array}\right)$

$m\mathbf{a}+n\mathbf{b}=\mathbf{\text{c}}$  

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.

$$\begin{gathered} \overrightarrow{OP}=\mathbf{a} \\ \overrightarrow{PT}=\mathbf{b} \\ \overrightarrow{OR}=3\mathbf{b} \end{gathered}$$

i) Find $\overrightarrow{OT}$ in terms of $\mathbf{a}$ and$\mathbf{b}$

[1]

ii) Find $\overrightarrow{PR}$ in terms of $\mathbf{a}$ and $\mathbf{b}$ Give your answer in its simplest form. [1]

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

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

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

$\overrightarrow{AB}=\left(\begin{array}{c}-3\\ 5\end{array}\right)$

i) Calculate $\left|\overrightarrow{AB}\right|$.

$\left|\overrightarrow{AB}\right|$ = ................................................ [2]

ii) $\overrightarrow{AC}=\left(\begin{array}{c}6\\ 2\end{array}\right)$ and C is the point (10, -1).

a) Find the coordinates of the point $A$.

[1]

b) $B$ is the midpoint of $AD$.  Find the coordinates of the point $D$. [2]

The diagram shows triangle $OPQ$.
 $OP=\mathbf{p}$ and $OQ=\mathbf{q}$.  
$R$ is the point on $OQ$ such that $OR=2RQ$.  
$S$ is the midpoint of $PQ$.  
Express, as simply as possible, in terms of$\mathbf{p}$ and/or $\mathbf{q}$

i) $\overrightarrow{PQ}$,

$\overrightarrow{PQ}=$...................[1]

ii) $\overrightarrow{OS}$

$\overrightarrow{OS}$=........................[2]

iii) $\overrightarrow{SR}$

$\overrightarrow{SR}$=...................[2]

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 $AC$.
$\overrightarrow{AC}=\mathbf{a}$and $\overrightarrow{CB}=\mathbf{b}$.  

Find, in terms of $\mathbf{a}$ and $\mathbf{b}$, in its simplest form 

i) $\overrightarrow{AB}$,

     $\overrightarrow{AB}=$................................................ [1]

ii) $\overrightarrow{OE}$,

     $\overrightarrow{OE}=$................................................ [2]

iii) the position vector of $D$.

      [3]

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

Find, in terms of $\mathbf{p}$ and/or $\mathbf{r}$, in its simplest form

i) $\overrightarrow{MQ}$,

$\overrightarrow{MQ}=$ ..............................................  [1]

ii) $\overrightarrow{MT}$,

$\overrightarrow{MT}=$ .............................................. [1]

iii) $\overrightarrow{OT}$,

$\overrightarrow{OT}=$ .............................................. [1]

$RQ$and $OT$are extended and meet at $U$.

Find the position vector of $U$ in terms of $\mathbf{p}$ and $\mathbf{r}$. 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. $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OC}=\mathbf{c}$.  

Find the following vectors, in terms of $\mathbf{a}$ and $\mathbf{c}$, in their simplest form

i) $\overrightarrow{AB}$,

$\overrightarrow{AB}=$................................................ [1]

ii) $\overrightarrow{AD}$,

$\overrightarrow{AD}=$................................................ [1]

iii) $\overrightarrow{CE}$,

$\overrightarrow{CE}=$................................................ [1]

iv) $\overrightarrow{CD}$.

$\overrightarrow{CD}=$................................................ [2]

$\overrightarrow{CE}=k\overrightarrow{CD}$

Find the value of $k$.

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

$ABC$is a straight line.

$AB:BC=2:5$

$$\begin{gathered} \overrightarrow{OA}=2\mathbf{a}+\mathbf{b} \\ \overrightarrow{OB}=3\mathbf{a}+2\mathbf{b} \end{gathered}$$

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