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Vectors (Cambridge O Level Maths)

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Vectors

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1
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stack V W with rightwards arrow on top equals open parentheses table row 10 row cell negative 24 end cell end table close parentheses

Find  open vertical bar stack V W with rightwards arrow on top close vertical bar.

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2
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stack A B with rightwards arrow on top space equals space open parentheses 6 minus 1 close parentheses               stack B C with rightwards arrow on top space equals space open parentheses negative 2 5 close parentheses                stack D C with rightwards arrow on top space equals space open parentheses 2 minus 3 close parentheses

Find

i)
stack A C with rightwards arrow on top,
 
stack A C with rightwards arrow on top space equals space open parentheses space space space space space space space space space space close parentheses  [2]
ii)
stack B D with rightwards arrow on top,
 
stack B D with rightwards arrow on top space equals space open parentheses space space space space space space space space space space close parentheses  [2]
iii)
open vertical bar stack B C with rightwards arrow on top close vertical bar.
  
  [2]

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3
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bold p equals open parentheses table row 4 row 5 end table close parentheses       bold q equals open parentheses table row cell negative 2 end cell row 7 end table close parentheses

i)

Find 2 bold p plus bold q.
 

open parentheses table row cell space space space space space end cell row blank end table close parentheses [2]

ii)

Find open vertical bar bold p close vertical bar.
 

[2]

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4
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bold a equals open parentheses table row cell negative 3 end cell row 2 end table close parentheses space space space space space space space bold b equals open parentheses table row 5 row 4 end table close parentheses space space space space space space space bold c equals open parentheses table row 14 row 9 end table close parentheses

i)
Find  3 bold a minus 2 bold b .
 
 

 open parentheses table row cell space space space space space end cell row cell space space space space space end cell end table close parentheses [2]

ii)
Find  vertical line space bold a bold space vertical line.
 
 

[2]

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5
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cie-igcse-2018-oct-nov-p4-tz3-q1b

i)
Find the column vector stack A B with rightwards arrow on top.
 
 

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 space space end cell end table close parentheses [1]

ii)
Find  vertical line stack A B with rightwards arrow on top vertical line.
 
 

vertical line stack A B with rightwards arrow on top vertical line equals................................................. [2]

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6
Sme Calculator5 marks

stack O A with rightwards arrow on top equals open parentheses table row 4 row 3 end table close parentheses      stack A B with rightwards arrow on top equals open parentheses table row 8 row cell negative 7 end cell end table close parentheses      stack A C with rightwards arrow on top equals open parentheses table row cell negative 3 end cell row 6 end table close parentheses

Find

i)
open vertical bar stack O B with rightwards arrow on top close vertical bar,


open vertical bar stack O B with rightwards arrow on top close vertical bar equals............................................... [3]

ii)
stack B C with rightwards arrow on top.


stack B C with rightwards arrow on top equalsopen parentheses table row blank row blank end table space space space close parentheses [2] 

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7
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Find the magnitude of the vector open parentheses table row cell negative 1 end cell row 7 end table close parentheses.

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1a
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The diagram shows triangle O P T.

q22-paper4-spec-2025-cie-igcse-further-maths

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.

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

i)   stack P L with rightwards arrow on top

[2]

ii)  stack K L with rightwards arrow on top

[2]

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1b
Sme Calculator2 marks

K L is extended to the point M.

stack K M with rightwards arrow on top space equals space minus 2 over 3 bold t bold space plus space 4 over 3 bold p.

Show that M lies on O P extended.

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2
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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.

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

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3
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cie-igcse-2020-may-jun-p4-tz2-q2d

In the diagram, O is the origin, O T equals 2 T D and M is the midpoint of T C.
stack O C with rightwards arrow on top equals bold c and stack O D with rightwards arrow on top equals bold d.
 
Find the position vector of M.
Give your answer in terms of bold c and bold d in its simplest form.

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4
Sme Calculator5 marks

bold a equals open parentheses table row cell negative 3 end cell row 2 end table close parentheses space space space space space space space bold b equals open parentheses table row 5 row 4 end table close parentheses space space space space space space space bold c equals open parentheses table row 14 row 9 end table close parentheses

m bold a plus n bold b equals text bold c end text
 
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 = ................................................      

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5a
Sme Calculator2 marks

q2a-hard-4-13-vectors-edexcel-gcse-maths

O P T R spaceis a trapezium.

stack O P with rightwards arrow on top equals bold a space space stack P T with rightwards arrow on top equals bold b space stack O R with rightwards arrow on top space equals space 3 bold b

i)
Find stack O T with rightwards arrow on top in terms of bold a andbold space bold b
[1]
ii)
Find stack P R with rightwards arrow on top in terms of bold a and bold b
Give your answer in its simplest form.
[1]

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5b
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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.

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5c
Sme Calculator2 marks

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

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6a
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stack A B with rightwards arrow on top equals open parentheses table row cell negative 3 end cell row 5 end table close parentheses

i)
Calculate open vertical bar stack A B with rightwards arrow on top close vertical bar.
open vertical bar stack A B with rightwards arrow on top close vertical bar = ................................................ [2]
ii)
stack A C with rightwards arrow on top equals open parentheses table row 6 row 2 end table close parentheses and C is the point (10, -1).
a)
Find the coordinates of the point A.

[1]

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

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6b
Sme Calculator5 marks

q10b-june-2021-4024-21-cambridge-o-level-maths-d

The diagram shows triangle O P Q.
 O P space equals space bold p and O Q space equals bold space bold q.
 R is the point on O Q such that O R space equals space 2 R Q.
 S is the midpoint of P Q.
 Express, as simply as possible, in terms ofbold space bold p and/or bold q

i)
stack P Q with rightwards arrow on top,
stack P Q space with rightwards arrow on top equals...................[1]
ii)
stack O S with rightwards arrow on top
stack O S with rightwards arrow on top=........................[2]
iii)
stack S R with rightwards arrow on top
stack S R with rightwards arrow on top=...................[2]

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1
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cie-igcse-2020-oct-nov-p4-tz3-q8b

In the diagram, O A B and O E D are straight lines.
O is the origin, A is the midpoint of O B and E is the midpoint of A C.
stack A C with rightwards arrow on top space equals space bold a spaceand stack C B with rightwards arrow on top space equals space bold b.
 
Find, in terms of bold a and bold b, in its simplest form 

i)
stack A B with rightwards arrow on top,
    
stack space A B with rightwards arrow on top space equals space................................................ [1]
ii)
stack O E with rightwards arrow on top,
    
stack space O E with rightwards arrow on top space equals space................................................ [2]
iii)
the position vector of D.
    
 [3]

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2a
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cie-igcse-spec-p4-q6a-diagram

O P Q R is a rectangle and O is the origin.
M is the midpoint of R Q and P T space colon space T Q space equals space 2 space colon space 1.
stack O P with rightwards arrow on top space equals space bold p and  stack O R with rightwards arrow on top space equals space bold r.

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

i)
stack M Q with rightwards arrow on top,

stack M Q with rightwards arrow on top space equals ..............................................  [1]
ii)
stack M T with rightwards arrow on top,

stack M T with rightwards arrow on top space equals .............................................. [1]
iii)
stack O T with rightwards arrow on top,

stack O T with rightwards arrow on top space equals .............................................. [1]

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2b
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R Q spaceand space O T spaceare extended and meet at U.

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

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3a
Sme Calculator5 marks

cie-igcse-2018-oct-nov-p4-tz2-q11b

O A B is a triangle and C is the mid-point of O B.
D is on A B such that A D : D B = 3 : 5.
O A E is a straight line such that O A colon A E = 2 : 3.
stack O A with rightwards arrow on top equals bold a and stack O C with rightwards arrow on top equals bold c.
 
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]

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3b
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stack C E with rightwards arrow on top equals k stack C D with rightwards arrow on top

Find the value of k.

 

 

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

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4
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q2-vhard-4-13-vectors-edexcel-gcse-maths

A B C spaceis a straight line.

A B space colon thin space B C space equals space 2 colon 5

stack O A space with rightwards arrow on top equals space 2 bold a plus bold b stack O B with rightwards arrow on top space equals 3 bold a space plus 2 bold b

Express stack space O C space with rightwards arrow on top in terms of space bold a bold spaceand bold b.
Give your answer in its simplest form.

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