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

$\overrightarrow{GH}=\frac{5}{6}\left(2\mathbf{p}+\mathbf{q}\right)$            $\overrightarrow{JK}=\frac{5}{18}\left(2\mathbf{p}+\mathbf{q}\right)$

Write down two facts about the geometrical relationship between the vectors $\overrightarrow{GH}$ and $\overrightarrow{JK}$.

a parallelogram.
$\overrightarrow{OA}=\mathbf{p}$ and $\overrightarrow{OC}=\mathbf{q}.$
$E$ is the point on $AB$ such that $AE:EB=3:1.$

Find $\overrightarrow{OE}$, in terms of $\mathbf{p}$ and $\mathbf{q}$, in its simplest form.

$\overrightarrow{OE}=$ .................................................. 

$\overrightarrow{XY}=3\mathbf{a}+2\mathbf{b}$ and $\overrightarrow{ZY}=6\mathbf{a}+4\mathbf{b}$.

Write down two statements about the relationship between the points $X$, $Y$ and $Z$.

1 ......................................     
2 ......................................     

$ABCD$ is a parallelogram.
$N$ is the point on $BD$ such that $BN:ND=4:1$. $\overrightarrow{AB}=\mathbf{\text{s}}$ and $\overrightarrow{AD}=\mathbf{\text{t}}$.

Find, in terms of $\mathbf{s}$ and $\mathbf{t}$, an expression in its simplest form for $\overrightarrow{BD}$.  

$\overrightarrow{BD}$ = ....................................................

Find, in terms of $\mathbf{s}$ and $\mathbf{t}$, an expression in its simplest form for $\overrightarrow{CN}$.  

$\overrightarrow{CN}$ = ....................................................

$O$ is the origin, $\overrightarrow{OP}=\mathbf{p}$ and $\overrightarrow{OQ}=\mathbf{q}$. $QT:TP=2:1$

Find the position vector of $T$.
Give your answer in terms of $\mathbf{p}$ and $\mathbf{q}$, in its simplest form.

OAB is a triangle and ABC and PQC are straight lines. P is the midpoint of OA, Q is the midpoint of PC and OQ : QB = 3 : 1.

$\overrightarrow{OA}=4\mathbf{a}$ and $\overrightarrow{OB}=8\mathbf{b}$.

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

i) $\overrightarrow{AB}$,

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

ii) $\overrightarrow{OQ}$,

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

iii) $\overrightarrow{PQ}$,

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

By using vectors, find the ratio $AB:BC$.

......................... : ........................

$PQRS$ is a parallelogram with diagonals $PR$ and $PR$ intersecting at $X$.
$\overrightarrow{PQ}=\mathbf{a}$ and $\overrightarrow{PS}=\mathbf{b}$.

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

$\overrightarrow{QX}=$ ...............................................

$PQRS$ is a parallelogram.
$\overrightarrow{PQ}=3\mathbf{a}$ and $\overrightarrow{PS}=2\mathbf{b}$.

$T$ is a point on $QS$ such that $QT:TS=1:2$.

Find $\overrightarrow{QS}$, giving your answer as simply as possible in terms of $\mathbf{a}$ and $\mathbf{b}$.

Find $\overrightarrow{TR}$, giving your answer as simply as possible n terms of $\mathbf{a}$ and $\mathbf{b}$.

The diagram shows a parallelogram $CDEF$.  $\overrightarrow{FE}=\mathbf{m}$and $\overrightarrow{CE}=\mathbf{n}$.
$B$ is the midpoint of $CD$.
$FA=2AC$

Find an expression, in terms of $\mathbf{m}$and $\mathbf{n}$, for $\overrightarrow{AB}$. Give your answer in its simplest form.

$\overrightarrow{AB}=$ .................................................

The diagram shows a triangle $OAB$ and a straight line $OAC$.
$OA:OC=2:5$ and $M$ is the midpoint of $AB$. $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$.

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

$\overrightarrow{AB}=$.................................................

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

$\overrightarrow{MC}=$.................................................

$O$ is the origin and $OPQR$ is a parallelogram.
$SOP$ is a straight line with $SO=OP$.
$TRQ$ is a straight line with $TR=RQ$.
$STV$ is a straight line and $ST:TV=2:1$.
$\overrightarrow{OR}=\mathbf{a}$ and $\overrightarrow{OP}=\mathbf{b}$.

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

i) the position vector of $T$,  

[2]

ii) $\overrightarrow{RV}$.  

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

Show that $PT$ is parallel to $RV$.

$O$ is the origin, $\overrightarrow{OP}=2\overrightarrow{OA}$, $\overrightarrow{OQ}=3\overrightarrow{OB}$ and  $\overrightarrow{PM}=\overrightarrow{MQ}$.

$\overrightarrow{OP}=\mathit{p}$ and  $\overrightarrow{OQ}=\mathit{q}$.

Find, in terms of $\mathit{p}$ and $\mathit{q}$, in its simplest form .

$\overrightarrow{BA}=$...................................................

Find, in terms of $\mathit{p}$ and $\mathit{q}$, in its simplest form the position vector of $M$.

$OABC$ is a parallelogram and $O$ is the origin. $CK=2KB$ and $AL=LB$.
$M$ is the midpoint of $KL$.
$\overrightarrow{OA}=\mathbf{\text{p}}$ and $\overrightarrow{OC}=\mathbf{\text{q }}\text{.}$   

Find $\overrightarrow{KL}$ in terms of p and q, giving your answer in its simplest form. 

$\overrightarrow{KL}=$ ............................................

Find the position vector of $M$ in terms of p and q, giving your answer in its simplest form. 

$ABCD$ is a parallelogram with $\overrightarrow{AB}=\mathbf{\text{q}}$ and $\overrightarrow{AD}=\mathbf{\text{p}}$
$ABM$ is a straight line with $AB:BM=1:1$.
$ADN$ is a straight line with $AD:DN=3:2$.

Write $\overrightarrow{MN}$, in terms of $\mathbf{\text{p}}$ and $\mathbf{\text{q}}$, in its simplest form.   

$\overrightarrow{MN}=$ ..............................................

The straight line $NM$ cuts $BC$ at $X.$
$X$ is the midpoint of $MN$.
$\overrightarrow{BX}=k\mathbf{\text{p}}$

Find the value of $k$.   

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

In the diagram, $OABC$ is a parallelogram.
$OP$ and $CA$intersect at $X$ and $CP:PB=2:1$. $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OC}=\mathbf{c}$.

Find $\overrightarrow{OP}$, in terms of $\mathbf{a}$ and $\mathbf{c}$, in its simplest form.  

$\overrightarrow{OP}=$ ................................................

$CX:XA=2:3$

i) Find $\overrightarrow{OX}$, in terms of $\mathbf{a}$ and $\mathbf{c}$, in its simplest form. 

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

ii) Find $OX:XP$. 

$OX:XP=$................... : ................... [2]

$$\begin{gathered} \overrightarrow{MT}=\left(2k \\ -k\right) \end{gathered}$$  and  $\left|\overrightarrow{MT}\right|=\sqrt{180}$.  

Find the positive value of $k$.

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