Vectors (Edexcel IGCSE Maths): Exam Questions

The vector $a$and the vector $b$ are shown on the grid.

On the grid, draw and label vector $-2a$

Work out $a+2b$ as a column vector.

Here are two vectors.

$\vec{AB} = ( \begin{gathered} 5 \\ 3 \end{gathered} )$ $\vec{CB} = ( \begin{gathered} - 2 \\ 4 \end{gathered} )$

Find, as a column vector, $\overset{\rightarrow}{AC}$

Here are two vectors.

$\vec{ABspace} = ( \begin{gathered} 6 \\ - 9 \end{gathered} )$ $\vec{CB} = ( \begin{gathered} 1 \\ 3 \end{gathered} )$

Find the magnitude of $\overset{\rightarrow}{AC}$.

Vectors $a$ and 4 b are drawn on the grid.

Write vector $a$ as a column vector.

Find vector b as a column vector.

Vector $a = ( \begin{gathered} 3 \\ - 1 \end{gathered} )$ and vector $b = ( \begin{gathered} 1 \\ 3 \end{gathered} )$

Gavin starts to draw a diagram to show that $a + 2 b = ( \begin{gathered} 5 \\ 5 \end{gathered} )$

Complete Gavin’s diagram.

Here is vector $a$.

Circle the column vector that represents $a$.

$a = ( \begin{gathered} - 3 \\ 2 \end{gathered} )$ and $b = ( \begin{gathered} 1 \\ - 5 \end{gathered} )$

Work out $a--3b$ Circle your answer.

Vector $a = ( \begin{gathered} 2 \\ 1 \end{gathered} )$ , vector $b = ( \begin{gathered} - 2 \\ 1 \end{gathered} )$.

On each grid below, draw a vector to represent

Emma says that if she draws vector $a$ and vector $b$ they will be the same.

Explain why this is incorrect.

Vectors a, b, c, d and e are drawn on an isometric grid.

Write each of the vectors c, d and e in terms of a and/or b.

c = ....................

d = ....................

e = ....................

$OAB$is a triangle.

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

Find $\overset{\rightarrow}{AB}$ in terms of $a$and $b$.

$P$ is the point on $AB$ such that $AP:PB=3:1$

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

$PQRS$is a parallelogram. $N$ is the point on $SQ$such that $SN:NQ=3:2$

$\overset{\rightarrow}{PQ}=a\,\overset{\rightarrow}{PS}=b$

Write down, in terms of $a$and $b$, an expression for $\overset{\rightarrow}{SQ}.$

Express $\overset{\rightarrow}{NR}$ in terms of $a$ and$b$.

$OAB$ is a triangle. $M$ is the midpoint of $OA$. $N$ is the midpoint of $OB$.

$\overset{\rightarrow}{OM}=m\,\overset{\rightarrow}{ON}=n$

Show that $AB$is parallel to $MN$.

$OAYB$ is a quadrilateral.

$\overset{\rightarrow}{OA}=3a\,\overset{\rightarrow}{OB}=6b$

Express $\overset{\rightarrow}{AB}$ in terms of $a$and $b$.

$X$ is the point on$AB$such that $AX:XB=1:2$ and $\overset{\rightarrow}{BY}=5a-b$

Prove that $\overset{\rightarrow}{OX}=\frac{2}{5}\overset{\rightarrow}{OY}$

$ABCDEF$ and $GHIJKL$ are regular hexagons each with centre $O$.

$GHIJKL$ is an enlargement of $ABCDEF$, with centre $O$ and scale factor 2

$\overset{\rightarrow}{OA}=a\overset{\rightarrow}{OB}=b$

Write the following vectors, in terms of $a$and $b$. Simplify your answers.

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

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

iii) $\overset{\rightarrow}{LD}$

$ABCDEF$ is a regular hexagon.

$ABX$ and $DCX$ are straight lines.

$\underset{AB}{\rightarrow}=a\underset{BC}{\rightarrow}=b$

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

$b$ is a vector.

Given that $b + ( \begin{gathered} 5 \\ 2 \end{gathered} )$ is parallel to $( \begin{gathered} 2 \\ 1 \end{gathered} )$ , find two possible answers for $b$.

$b = ( \begin{gathered} \\ \end{gathered} )$ or $( \begin{gathered} \\ \end{gathered} )$

$OPTR$is a trapezium.

$\overset{\rightarrow}{OP}=a\,\overset{\rightarrow}{PT}=b\,\overset{\rightarrow}{OR}=4b$

(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.

$APB$ is a triangle. $N$ is a point on $AP$.

$\overset{\rightarrow}{AB}=a\overset{\rightarrow}{AN}=2b\overset{\rightarrow}{NP}=b$

Find the vector $\overset{\rightarrow}{PB}$, in terms of $a$ and $b$.

$B$ is the midpoint of $AC$. $M$ is the midpoint of $PB$.

Show that $NMC$ is a straight line.

$OPTR$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.

$S$ 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.

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

$OAB$is a triangle. $\overset{\rightarrow}{OA}=5a\,\overset{\rightarrow}{OB}=2b$

$T$ is the point on $AB$such that $AT:TB=5:1$

Show that $OT$is parallel to the vector $a+2b$

$\overset{\rightarrow}{OA}=6a$ and $\overset{\rightarrow}{OB}=6b$ $M$ is the midpoint of $AB$.

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

$N$ is the midpoint of $OB$. $G$ is the point on $OM$ such that $OG:GM$= 2 : 1

Show that $AGN$ is a straight line.

$CAYB$ is a quadrilateral.

$\overset{\rightarrow}{CA}=3a\,\overset{\rightarrow}{CB}=6b\,\overset{\rightarrow}{BY}=5a-b$

$X$ is the point on $AB$such that $AX:XB=1:2$

Prove that $\overset{\rightarrow}{CX}=\frac{2}{5}\overset{\rightarrow}{CY}$

$OAB$is a triangle. $P$ is the point on $AB$such that $AP:PB=5:3$

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

$\overset{\rightarrow}{OP}=k(3a+5b)$ where $k$ is a scalar quantity.

Find the value of $k$.

$ABC$is a triangle. The midpoint of $BC$is $M$. $P$ is a point on $AM$.

$\begin{gathered} \vec{AB} & = 4 a \\ \vec{AC} & = 2 b \\ \vec{AP} & = \frac{3}{2} a + \frac{3}{4} b \end{gathered}$

Find the ratio $AP:PM$

$OAB$ is a triangle. $\underset{OA}{\rightarrow}=a\underset{OB}{\rightarrow}=b$

The point $C$ lies on $OA$ such that $OC:CA=1:2$ The point $D$ lies on $OB$ such that $OD:DB=1:2$

Using a vector method, prove that $ABDC$ is a trapezium.

The diagram shows parallelogram $ABCD$.

$\vec{AB} = ( \begin{gathered} 2 \\ 7 \end{gathered} )$ $\vec{AC} = ( \begin{gathered} 10 \\ 11 \end{gathered} )$

The point $B$ has coordinates (5, 8) Work out the coordinates of the point $C$.

The point $E$ has coordinates (63, 211) Use a vector method to prove that $ABE$ is a straight line.

The diagram shows triangle $ABD.$

$N$ is the midpoint of $BC$.

$C$ is the midpoint of $AD$.

$M$ is the point on $AB$ such that $AM:MB=3:1$

$\overset{\rightarrow}{AB}=p$ and $\overset{\rightarrow}{AC}=q$

Express, in terms of $p$ and $q$,

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

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

State, giving reasons, two different geometric facts relating $MN$ and $BD.$

In triangle $JKL$

$M$ is the midpoint of $JK$ $JN:NL=3:2$ $\overset{\rightarrow}{KL}=7a$ $\overset{\rightarrow}{NL}=4b$

Work out $\overset{\rightarrow}{JM}$ in terms of $a$ and $b$.

Give your answer in its simplest form.

$PQR$ is a straight line.

$PQ:QR=3:1$

$\overset{\rightarrow}{PQ}=a$

Circle the vector $\overset{\rightarrow}{RQ}$

$a = ( \begin{gathered} 6 \\ - 10 \end{gathered} )$ $b = ( \begin{gathered} - 1 \\ 2 \end{gathered} )$ $c = ( \begin{gathered} - 4 \\ 7 \end{gathered} )$

Work out $a+b+c$

Show that $a+2c$ is parallel to $b$

Is $BCD$ a straight line? Show working to support your answer.

Given that

$m ( \begin{gathered} 4 \\ 1 \end{gathered} ) + n ( \begin{gathered} 5 \\ 2 \end{gathered} ) = ( \begin{gathered} 12 \\ 6 \end{gathered} )$

find the value of $m$ and the value of $n$.

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

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

Two vectors, a and b, are shown on the 1 centimetre grid below.

Show that the vector a + 2b has length 7cm. You may use the grid below.

Vector $a = ( \begin{gathered} 2 \\ 1 \end{gathered} )$ , vector $b = ( \begin{gathered} - 2 \\ 1 \end{gathered} )$.

$c = ( \begin{gathered} - 12 \\ 0 \end{gathered} )$

Find the value $k$ so that $k ( a - b ) = c$

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

P, Q, R and S are the midpoints of OX, XY, YZ and OZ respectively.

$\overset{\rightarrow}{\mathrm{OP}}=a$ , $\overset{\rightarrow}{\mathrm{XQ}}=b$ and $\overset{\rightarrow}{\mathrm{OS}}=c$.

Show that $\mathrm{PQ}$ is parallel to $\mathrm{SR}$.

The diagram shows triangle $ABD$.

$N$ is the midpoint of $BC$. $C$ is the midpoint of $AD$. $M$ is the point on $AB$ such that $AM:MB=3:1$

$\overset{\rightarrow}{AB}=p$ and $\overset{\rightarrow}{AC}=q$

Express, in terms of $p$ and $q$,

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

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

State, giving reasons, two different geometric facts relating $MN$ and $BD$

$OMA,ONB,MPB$and $NPA$are straight lines.

$M$ is the midpoint of $OA$

$ON:NB=1:5$

$\overset{\rightarrow}{OM}=a\overset{\rightarrow}{ON}=b$

Find in terms of $a$ and $b$the vector $\overset{\rightarrow}{AN}$

Use a vector method to find the ratio $AP:PN$

$OCAB$ is a quadrilateral.

$\overset{\rightarrow}{OA}=3a\overset{\rightarrow}{OB}=4b\overset{\rightarrow}{BC}=2a-2b$

(i) Find the vector $\overset{\rightarrow}{OC}$ in terms of a and b Simplify your answer.

(ii) Find the vector $\overset{\rightarrow}{AB}$ in terms of a and b

The point $P$ lies on $AB$ and on $OC$

Using a vector method, find the ratio $AP:PB$ Show your working clearly.

$OACB$ is a parallelogram.

$\overset{\rightarrow}{OA}=a\mathrm{and}\overset{\rightarrow}{OB}=b$

$D$is the point such that $\overset{\rightarrow}{AC}=\overset{\rightarrow}{CD}$ The point $N$ divides $AB$in the ratio $2:1$

Write an expression for $\overset{\rightarrow}{ON}$ in terms of $a$and $b$.

Prove that $OND$is a straight line.

$ABC$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.

$ACEF$is a parallelogram. $B$ is the midpoint of $AC$. $M$ is the midpoint of $BE$.

$\overset{\rightarrow}{CB}=a\,\overset{\rightarrow}{ED}=b\,\overset{\rightarrow}{DC}=2b$

Show that $AMD$ is a straight line.

$OABC$is a parallelogram. $\overset{\rightarrow}{OA}=a\mathrm{and}\overset{\rightarrow}{OC}=c$

$X$ is the midpoint of the line $AC$.

$OCD$is a straight line so that $OC:CD=k:1$

Given that $\overset{\rightarrow}{XD}=3c-\frac{1}{2}a$

find the value of$k.$

$OAN,OMB$ and $APB$are straight lines. $AN=2OA.$ $M$ is the midpoint of $OB$.

$\overset{\rightarrow}{OA}=a\overset{\rightarrow}{OB}=b$

$\overset{\rightarrow}{AP}=k\overset{\rightarrow}{AB}$ where $k$ is a scalar quantity.

Given that $MPN$is a straight line, find the value of $k$.

$OMA,ONB$ and $ABC$are straight lines. $M$ is the midpoint of $OA$. $B$ is the midpoint of $AC$.

$\overset{\rightarrow}{OA}=6a\overset{\rightarrow}{OB}=6b\overset{\rightarrow}{ON}=kb$ where $k$ is a scalar quantity.

Given that $MNC$is a straight line, find the value of $k$.

$CDEF$is a quadrilateral.

$\overset{\rightarrow}{CD}=a,\overset{\rightarrow}{DE}=b\mathrm{and}\overset{\rightarrow}{FC}=a-b.$

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

$M$ is the midpoint of $DE$. $X$ is the point on $FM$such that $FX:XM=n:1$ $CXE$ is a straight line.

Work out the value of $n$.

The diagram shows triangle $OAB$

$\overset{\rightarrow}{OA}=8a$ $\overset{\rightarrow}{OB}=6b$

$M$ is the point on $OB$ such that $OM:MB=1:2$ $N$ is the midpoint of $AB$ $P$ is the point of intersection of $ON$ and $AM$

Using a vector method, find $\overset{\rightarrow}{OP}$ as a simplified expression in terms of $a$ and $b$ Show your working clearly.

$\overset{\rightarrow}{OP}$= ...................................

$OXYZ$ is a trapezium.

$\overset{\rightarrow}{OX}=a$

$\overset{\rightarrow}{XY}=b$

$\overset{\rightarrow}{OZ}=3b$

$M$is the midpoint of $OX$ $N$ is the point such that $MNZ$ and $ONY$ are straight lines.

Given that $ON:OY=λ:1$

use a vector method to find the value of $λ$

$λ=$ ....................................................

$OACB$is a trapezium.

$\overset{\rightarrow}{OA}=2a$ $\overset{\rightarrow}{OB}=5b$ $\overset{\rightarrow}{AC}=3b$

The diagonals, OC and AB, of the trapezium intersect at the point $P$.

Find and simplify an expression, in terms of $a$ and $b$, for $\overset{\rightarrow}{OP}$ Show your working clearly.

$\overset{\rightarrow}{OP}=$...............................

In the diagram

$\begin{gathered} \vec{DE} & = a \\ \vec{DH} & = b \\ \vec{HG} & = 8 b \\ E X : X H & = 3 : 1 \\ E F : F G & = 1 : 3 \\ \end{gathered}$

Show that $\overset{\rightarrow}{DX}=\frac{1}{4}a+\frac{3}{4}b$

Is $DXF$ a straight line?

Show working to support your answer.

Here is quadrilateral $PQRS$.

$\overset{\rightarrow}{PS}=a$ $\overset{\rightarrow}{SR}=b$ $\overset{\rightarrow}{RQ}=\text{c}$

$X$ is a point on $PS$ where $PX:XS=1:2$