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Vectors (Cambridge O Level Maths)

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Vectors

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Vectors

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1a
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bold u bold space equals space open parentheses table row 3 row cell negative 2 end cell end table close parentheses space space space space space space space space space space space bold italic v space equals space open parentheses table row cell negative 12 end cell row 5 end table close parentheses

Find bold u space – space 2 bold v.

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1b
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Find space vertical line bold v vertical line space .

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2
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Ahmed finds the magnitude of the vector open parentheses space space space space 2 minus 3 close parentheses.

From this list, select the correct calculation.
 

square root of 2 squared space plus space minus 3 squared end root space space space space space space space space space space 2 to the power of 2 space end exponent minus space 3 squared space space space space space space space space space space square root of 2 squared space minus space 3 squared end root space space space space space space space space space space 2 to the power of 2 space end exponent plus space open parentheses negative 3 squared close parentheses space space space space space space space space space space square root of 2 squared space plus space open parentheses negative 3 close parentheses squared end root

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3
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bold m equals open parentheses table row 5 row 7 end table close parentheses

Find 3 bold m.


open parentheses table row cell space space space space space space end cell row blank end table close parentheses  

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4
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Point A has coordinates left parenthesis 6 comma space 4 right parenthesis and point B has coordinates left parenthesis 2 comma space 7 right parenthesis.

Write stack A B with rightwards arrow on top as a column vector.


stack A B with rightwards arrow on top equals open parentheses table row cell space space space space end cell row cell space space space space end cell end table close parentheses 

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5
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O is the origin, stack O A with rightwards arrow on top equals 2 bold x plus 3 bold y and stack B A with rightwards arrow on top equals bold x minus 4 bold y.
 
Find the position vector of B, in terms of bold x and bold y, in its simplest form.

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6a
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cie-igcse-2018-may-jun-2-p2-q22
In the diagram, O is the origin, stack O C with rightwards arrow on top equals negative 2 bold a plus 3 bold b and stack O D with rightwards arrow on top equals 4 bold a plus bold b.

Find stack C D with rightwards arrow on top, in terms of bold a and bold b, in its simplest form.

stack C D with rightwards arrow on top ................................................

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6b
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stack D E with rightwards arrow on top equals bold a minus 2 bold b

Find the position vector of E, in terms of bold a and bold b, in its simplest form.

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7
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cie-igcse-2018-march-p2-7

The diagram shows a regular hexagon A B C D E F.
stack C D with rightwards arrow on top equals bold p and stack C B with rightwards arrow on top equals bold q.

Find stack C A with rightwards arrow on top, in terms of bold p and bold q, giving your answer in its simplest form.


stack C A with rightwards arrow on top= .......................................

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8
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A is the point left parenthesis 4 comma space 1 right parenthesis and stack A B with rightwards arrow on top equals open parentheses table row cell negative 3 end cell row 1 end table close parentheses.
 
Find the coordinates of B.
 

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

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9
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cie-igcse-2019-oct-nov-p4-tz2-q8b

i)table row blank row blank end table
Write stack O A with rightwards arrow on top as a column vector.

 

stack O A with rightwards arrow on top equals open parentheses table row cell space space space space space space space space end cell row blank end table close parentheses  [1]

ii)table row blank row blank end table
Write stack A B with rightwards arrow on top as a column vector.

 

stack A B with rightwards arrow on top equals open parentheses table row cell space space space space space space space space end cell row blank end table close parentheses  [1]

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10
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The position vector of point A is open parentheses table row cell negative 2 end cell row 3 end table close parentheses.

stack A B with rightwards arrow on top equals open parentheses table row 5 row cell negative 1 end cell end table close parentheses and  stack A B with rightwards arrow on top equals stack B C with rightwards arrow on top.

Find the position vector of point C.

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11a
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q20-specimen-paper1-2025-4024-01-cambridge-o-level-maths-d

O A C B is a parallelogram.
stack space O A with rightwards arrow on top space equals space 2 bold f spaceand stack O B with rightwards arrow on top space equals space 2 bold g.

X is a point on A B such that A X colon X B space equals space 3 colon space 1.
Find, as simply as possible, in terms of bold f and bold g.

stack A B with rightwards arrow on top

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11b
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Find, as simply as possible, in terms of bold f and bold g.

stack X C with rightwards arrow on top.

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12a
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q7-june-2018-4024-1-cambridge-o-level-maths-d

A B C D spaceis a parallelogram.
X is the point on B C such that space B X space colon space X C space equals space 2 space colon space 1. space

stack A B with rightwards arrow on topequals space 2 bold italic p and stack A D with rightwards arrow on top space equals space 3 bold italic q.
Find, in terms of bold italic p and bold italic q,
 

stack A C with rightwards arrow on top ,

stack A C with rightwards arrow on top = ..................................

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12b
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stack A X with rightwards arrow on top,

stack A X with rightwards arrow on top..............................

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12c
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stack X D with rightwards arrow on top.

stack X D with rightwards arrow on top..........................

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1
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stack G H with rightwards arrow on top space equals 5 over 6 open parentheses 2 bold p space plus bold space bold q close parentheses            stack J K with rightwards arrow on top space equals 5 over 18 open parentheses 2 bold p space plus bold space bold q close parentheses

Write down two facts about the geometrical relationship between the vectors stack G H with rightwards arrow on top and stack J K with rightwards arrow on top.

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2
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cie-igcse-2020-may-june-p2-tz1-q17

a parallelogram.
stack O A with rightwards arrow on top equals bold p and stack O C with rightwards arrow on top equals bold q.
E is the point on A B such that A E space colon space E B space equals space 3 space colon space 1.

Find stack O E with rightwards arrow on top, in terms of bold p and bold q, in its simplest form.


stack O E with rightwards arrow on top equals .................................................. 

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3
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stack X Y with rightwards arrow on top equals 3 bold a plus 2 bold b and stack Z Y with rightwards arrow on top equals 6 bold a plus 4 bold b.

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


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

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4a
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cie-igcse-2019-may-jun-p2-tz3-q22a

A B C D is a parallelogram.
N is the point on B D such that B N colon N D equals 4 colon 1.
stack A B with rightwards arrow on top equals text bold s end text and stack A D with rightwards arrow on top equals text bold t end text.

Find, in terms of bold s and bold t, an expression in its simplest form for stack B D with rightwards arrow on top.
 

stack B D with rightwards arrow on top = ....................................................

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4b
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Find, in terms of bold s and bold t, an expression in its simplest form for stack C N with rightwards arrow on top.
 

stack C N with rightwards arrow on top = ....................................................

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5
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cie-igcse-2018-may-jun-3-p2-q14

O is the origin, stack O P with rightwards arrow on top equals bold p and stack O Q with rightwards arrow on top equals bold q.
Q T space colon space T P space equals space 2 space colon space 1

Find the position vector of T.
Give your answer in terms of bold p and bold q, in its simplest form.

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6a
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screenshot-2022-09-15-at-10-34-46-am

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.

stack O A with rightwards arrow on top equals 4 bold a and stack O B with rightwards arrow on top equals 8 bold b.

Find, in terms of bold a and/or bold b, in its simplest form

i)
stack A B with rightwards arrow on top,

 

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

ii)
stack O Q with rightwards arrow on top,

 

stack O Q with rightwards arrow on top = ................................................... [1]

iii)
stack P Q with rightwards arrow on top,

 

stack P Q with rightwards arrow on top = ................................................... [1]

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6b
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By using vectors, find the ratio A B space colon space B C.

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

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7
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cie-igcse-2018-may-jun-1-p4-q11b
P Q R S is a parallelogram with diagonals P R and P R intersecting at X.
stack P Q with rightwards arrow on top equals bold a and stack P S with rightwards arrow on top equals bold b.

Find stack Q X with rightwards arrow on top in terms of bold a and bold b.
Give your answer in its simplest form.

stack Q X with rightwards arrow on top equals ...............................................

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1
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cie-igcse-2020-oct-nov-p2-tz1-q23

The diagram shows a parallelogram C D E F.
 stack F E with rightwards arrow on top space equals bold space bold m italic spaceand stack C E with rightwards arrow on top space equals space bold n.
B is the midpoint of C D.
F A space equals space 2 A C

Find an expression, in terms of bold m bold spaceand bold n, for stack A B with rightwards arrow on top.
Give your answer in its simplest form.

stack A B with rightwards arrow on top space equals .................................................

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2a
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cie-igcse-2020-oct-nov-p2-tz2-q22a

The diagram shows a triangle O A B and a straight line O A C.
O A space colon space O C space equals space 2 space colon space 5 and M is the midpoint of A B.
stack O A with rightwards arrow on top space equals bold space bold a and stack O B with rightwards arrow on top space equals space bold b.

Find stack A B with rightwards arrow on top, in terms of bold a and bold b, in its simplest form.

stack A B with rightwards arrow on top space equals space.................................................

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2b
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Find stack M C with rightwards arrow on top, in terms of bold a and bold b, in its simplest form.

stack M C with rightwards arrow on top space equals space.................................................

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3a
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cie-igcse-2020-mary-jun-p2-tz3-q21

O is the origin and O P Q R is a parallelogram.
S O P is a straight line with S O equals O P.
T R Q is a straight line with T R equals R Q.
S T V is a straight line and S T space colon space T V equals 2 space colon space 1.
stack O R with rightwards arrow on top equals bold a and stack O P with rightwards arrow on top equals bold b.

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

i)

the position vector of T,
 

[2]

ii)

stack R V with rightwards arrow on top.
 

stack R V with rightwards arrow on top equals................................................ [1]

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3b
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Show that P T is parallel to R V.

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4a
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cie-igcse-2019-oct-nov-p2-tz1-q25

O is the origin, stack O P with rightwards arrow on top equals 2 stack O A with rightwards arrow on top, stack O Q with rightwards arrow on top equals 3 stack O B with rightwards arrow on top and  stack P M with rightwards arrow on top equals stack M Q with rightwards arrow on top.

stack O P with rightwards arrow on top equals bold italic p and  stack O Q with rightwards arrow on top equals bold italic q.

Find, in terms of bold italic p and bold italic q, in its simplest form .

stack B A with rightwards arrow on top equals...................................................

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4b
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Find, in terms of bold italic p and bold italic q, in its simplest form the position vector of M.

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5a
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cie-igcse-2019-may-june-1-q25

O A B C is a parallelogram and O is the origin.
C K equals 2 K B and A L equals L B.
M is the midpoint of K L.
stack O A with rightwards arrow on top equals text bold p end text and stack O C with rightwards arrow on top equals text bold q  end text text. end text 
 
Find stack K L with rightwards arrow on top in terms of p and q, giving your answer in its simplest form. 

 

stack K L with rightwards arrow on top equals ............................................

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5b
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Find the position vector of M in terms of p and q, giving your answer in its simplest form. 

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6a
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cie-igcse-2019-may-jun-p2-tz2-q23a

A B C D is a parallelogram with stack A B with rightwards arrow on top equals text bold q end text and stack A D with rightwards arrow on top equals text bold p end text.
A B M is a straight line with A B colon B M equals 1 colon 1.
A D N is a straight line with A D colon D N equals 3 colon 2.

Write stack M N with rightwards arrow on top, in terms of text bold p end text and text bold q end text, in its simplest form. 
 

stack M N with rightwards arrow on top equals ..............................................

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6b
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The straight line N M cuts B C at X.
X is the midpoint of M N.
stack B X with rightwards arrow on top equals k text bold p end text

Find the value of k. 
 

k = .............................................

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7a
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cie-igcse-2018-oct-nov-p2-tz3-q26

In the diagram, O A B C is a parallelogram.
O P and C Aintersect at X and C P space colon space P B space equals space 2 space colon space 1.
stack O A with rightwards arrow on top equals bold a and stack O C with rightwards arrow on top equals bold c.

Find stack O P with rightwards arrow on top, in terms of bold a and bold c, in its simplest form.
 

stack O P with rightwards arrow on top equals ................................................

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7b
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C X space colon space X A space equals space 2 space colon space 3

i)
Find stack O X with rightwards arrow on top, in terms of bold a and bold c, in its simplest form.
 
 

stack O X with rightwards arrow on top equals................................................ [2]

ii)
Find O X colon X P.
 
 

O X colon X P equals................... : ................... [2]

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8
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stack M T with rightwards arrow on top space space equals space open parentheses 2 k minus k close parentheses  and  open vertical bar stack M T with rightwards arrow on top space close vertical bar equals space square root of 180.
 
Find the positive value of k.


k space equals ..............................................

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