A star is made up of a regular hexagon, centre X, surrounded by 6 equilateral triangles.{O|A # a and {O|B # b.

(b) When | a | # 5, write down the value of

(i) | b |, [1]

(ii) | a 0 b |. [1]

0580/4, 0581/4 Jun/03

6

O

P

Q

T

S

R

F

E

DC

B

A

X

a

b

7

(a) Write the following vectors in terms of a and�or b, giving your answers in their simplest form.

(i) {O|S, [1]

(ii) {A|B, [1]

(iii) {C|D, [1]

(iv) {O|R, [2]

(v) {C|F. [2]

6

� UCLES 2004 0580/2, 0581/2 Jun/04

For

Examiner's

Use16

C B

O A

M

a

c

OABC is a parallelogram. = a, = c and M is the mid-point of CA.Find in terms of a and c

(a) ,

Answer(a) [1]

(b) ,

Answer(b) [1]

(c) .

Answer(c) [2]

4

© UCLES 2005 0580/04, 0581/04 Jun 05

5 C

D

E

O

A

B

cdNOT TO

SCALE

OABCDE is a regular hexagon. With O as origin the position vector of C is c and the position vector of D is d. (a) Find, in terms of c and d,

(i) , [1]

(ii) , [2]

(iii) the position vector of B. [2]

3

© UCLES 2010 0580/42/M/J/10 [Turn over

For

Examiner's

Use

2 (a) p = 3

2

and q = 6

3

.

(i) Find, as a single column vector, p + 2q.

Answer(a)(i)

[2] (ii) Calculate the value of | p + 2q |. Answer(a)(ii) [2]

(b)

In the diagram, CM = MV and OL = 2LV.

O is the origin. = c and = v. Find, in terms of c and v, in their simplest forms

(i) , Answer(b)(i) [2]

(ii) the position vector of M, Answer(b)(ii) [2]

(iii) . Answer(b)(iii) [2]

C

O VL

M

NOT TOSCALE

11

© UCLES 2006 0580/02, 0581/02 Jun 06

For

Examiner's

Use

23

V W

S R

U P

QT

O

a b

The origin O is the centre of the octagon PQRSTUVW.

= a and = b. (a) Write down in terms of a and b

(i) ,

Answer(a)(i) [1]

(ii) ,

Answer(a)(ii) [1]

(iii) ,

Answer(a)(iii) [2] (iv) the position vector of the point P.

Answer(a)(iv) [1] (b) In the diagram, 1 centimetre represents 1 unit.

Write down the value of a – b.

Answer(b) [1]

16

© UCLES 2011 0580/43/M/J/11

For

Examiner's

Use

10 (a)

D

A

C

Bp

q

L

N

M

NOT TOSCALE

ABCD is a parallelogram. L is the midpoint of DC, M is the midpoint of BC and N is the midpoint of LM.

= p and = q. (i) Find the following in terms of p and q, in their simplest form.

(a) Answer(a)(i)(a) = [1]

(b) Answer(a)(i)(b) = [2]

(c) Answer(a)(i)(c) = [2]

(ii) Explain why your answer for shows that the point N lies on the line AC. Answer(a)(ii) [1]

10

© UCLES 2012 0580/42/M/J/12

For

Examiner's

Use

7 (a) P is the point (2, 5) and =

− 2

3 .

Write down the co-ordinates of Q. Answer(a) ( , ) [1]

(b)

D

CE

M

B

AO

c

3a

NOT TOSCALE

O is the origin and OABC is a parallelogram. M is the midpoint of AB.

= c, = 3a and CE = 3

1CB.

OED is a straight line with OE : ED = 2 : 1 . Find in terms of a and c, in their simplest forms

(i) , Answer(b)(i) = [1]

(ii) the position vector of M,

Answer(b)(ii) [2]

(iii) , Answer(b)(iii) = [1]

(iv) . Answer(b)(iv) = [2]

(c) Write down two facts about the lines CD and OB.

Answer (c)

[2]

6

© UCLES 2010 0580/23/M/J/10

For

Examiner's

Use

15

G

O

H

N

g

h

NOT TOSCALE

In triangle OGH, the ratio GN : NH = 3 : 1.

= g and = h. Find the following in terms of g and h, giving your answers in their simplest form.

(a) Answer(a) = [1]

(b) Answer(b) = [2]

10

© UCLES 2011 0580/21/M/J/11

For

Examiner's

Use

For

Examiner's

Use

18

Q

P

M

X

S

R

NOT TOSCALE

In the diagram, PQS, PMR, MXS and QXR are straight lines.

PQ = 2 QS.

M is the midpoint of PR.

QX : XR = 1 : 3.

= q and = r.

(a) Find, in terms of q and r,

(i) ,

Answer(a)(i) = [1]

(ii) .

Answer(a)(ii) = [1]

(b) By finding , show that X is the midpoint of MS.

Answer (b)

[3]

19

0580/41/M/J/14© UCLES 2014 [Turn over

(b)

A

N

O

Y

C

Ba

b

NOT TOSCALE

OACB is a parallelogram. = a and = b. AN : NB = 2 : 3 and AY = 5

2 AC.

(i) Write each of the following in terms of a and/or b. Give your answers in their simplest form.

(a)

Answer(b)(i)(a) = ................................................ [2]

(b)

Answer(b)(i)(b) = ................................................ [2]

(ii) Write down two conclusions you can make about the line segments NY and BC.

Answer(b)(ii) ...............................................................................................................................

..................................................................................................................................................... [2]__________________________________________________________________________________________

© UCLES 2011 0580/22/M/J/11

16

C

O

B

Aa

c

P

MQ

NOT TOSCALE

O is the origin and OABC is a parallelogram. CP = PB and AQ = QB.

= a and = c . Find in terms of a and c, in their simplest form,

(a) ,

Answer(a) = [2]

(b) the position vector of M, where M is the midpoint of PQ. Answer(b) [2]

8

0580/43/M/J/14© UCLES 2014

5 (a)

5

4

3

2

1

01 2 3 4 5 6 7 8

x

y

A

B

(i) Write down the position vector of A.

Answer(a)(i) f p [1]

(ii) Find ì ì , the magnitude of .

Answer(a)(ii) ................................................ [2]

(b)

p

q

O

Q

S

P R

NOT TOSCALE

O is the origin, = p and = q. OP is extended to R so that OP = PR. OQ is extended to S so that OQ = QS.

(i) Write down in terms of p and q.

Answer(b)(i) = ................................................ [1]

(ii) PS and RQ intersect at M and RM = 2MQ.

Use vectors to fi nd the ratio PM : PS, showing all your working.

Answer(b)(ii) PM : PS = ....................... : ....................... [4]__________________________________________________________________________________________

10

© UCLES 2012 0580/21/M/J/12

For

Examiner's

Use

19

O

T

P

Q

RS

t

p O is the origin and OPQRST is a regular hexagon.

= p and = t. Find, in terms of p and t, in their simplest forms,

(a) , Answer(a) = [1]

(b) , Answer(b) = [2]

(c) the position vector of R.

Answer(c) [2]

9

© UCLES 2012 0580/23/M/J/12 [Turn over

For

Examiner's

Use

18 Q R

O P

MXq

p

NOT TOSCALE

O is the origin and OPRQ is a parallelogram. The position vectors of P and Q are p and q. X is on PR so that PX = 2XR. Find, in terms of p and q, in their simplest forms

(a) , Answer(a) =

[2] (b) the position vector of M, the midpoint of QX. Answer(b) [2]

0580/21/M/J/13© UCLES 2013

Answer t = ............................................... [3]_____________________________________________________________________________________

20R Q

S

Mr

pO P

NOT TOSCALE

OPQR is a parallelogram, with O the origin. M is the midpoint of PQ. OM and RQ are extended to meet at S. = p and = r.

(a) Find, in terms of p and r, in its simplest form,

(i) ,

Answer(a)(i) = ............................................... [1]

(ii) the position vector of S.

Answer(a)(ii) ............................................... [1]

(b) When = – 2

1 p + r , what can you write down about the position of T ?

Answer(b) ................................................................................................................................. [1]_____________________________________________________________________________________

11

0580/23/M/J/13© UCLES 2013 [Turn over

ForExaminer′s

Use

19C

O

E A

D B

c

b

OABCDE is a regular polygon.

(a) Write down the geometrical name for this polygon.

Answer(a) ............................................... [1]

(b) O is the origin. = b and = c.

Find, in terms of b and c, in their simplest form,

(i) ,

Answer(b)(i) = ............................................... [1]

(ii) ,

Answer(b)(ii) = ............................................... [2]

(iii) the position vector of E.

Answer(b)(iii) ............................................... [1]_____________________________________________________________________________________

Question 20 is printed on the next page.

0580/22/M/J/14© UCLES 2014 [Turn over

14R

O P

Q

Mr

p

NOT TOSCALE

OPQR is a trapezium with RQ parallel to OP and RQ = 2OP. O is the origin, = p and = r. M is the midpoint of PQ.

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

(a) ,

Answer(a) = ................................................ [1]

(b) , the position vector of M.

Answer(b) = ................................................ [2]__________________________________________________________________________________________

In the diagram and .# p and # q.

(a) Find in terms of p and q

(i) ,

Answer (a)(i)� # ........................... [2]

(ii) .

Answer (a)(ii)� #........................... [2]

(b) AQ and BP intersect at T..

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

Answer (b)� # ............................... [2]{Q|T

{Q|TBT = �13 ���BP

{B|P

{B|P

{A|Q

{A|Q

{O|Q{O|POB = �34 ���OQOA = �23 ���OP

0580/02/0581/02/O/N/03 [Turn over

9 ForExaminer’s

use

A

Q

P

B

NOT TOSCALE

OT

20

9

© UCLES 2008 0580/21/O/N/08 [Turn over

For

Examiner's

Use

17

A B

C

O

q

p

O is the origin. Vectors p and q are shown in the diagram. (a) Write down, in terms of p and q, in their simplest form (i) the position vector of the point A, Answer(a)(i) [1]

(ii) BC ,

Answer(a)(ii) [1]

(iii) BC − AC .

Answer(a)(iii) [2]

(b) If | p | = 2, write down the value of | AB |.

Answer(b) [1]

6

© UCLES 2009 0580/21/O/N/09

For

Examiner's

Use

15

M

O R

P Q

r

p

O is the origin and OPQR is a parallelogram whose diagonals intersect at M.

The vector OP is represented by p and the vector is represented by r.

(a) Write down a single vector which is represented by (i) p + r, Answer(a)(i) [1]

(ii) 2

1 p – 2

1 r.

Answer(a)(ii) [1] (b) On the diagram, mark with a cross (x) and label with the letter S the point with position vector

2

1 p + 4

3 r. [2]

8

© UCLES 2011 0580/21/O/N/11

For

Examiner's

Use

13 CA B D

O

ab

A and B have position vectors a and b relative to the origin O.

C is the midpoint of AB and B is the midpoint of AD.

Find, in terms of a and b, in their simplest form

(a) the position vector of C,

Answer(a) [2]

(b) the vector .

Answer(b) [2]

11

© UCLES 2011 0580/22/O/N/11 [Turn over

For

Examiner's

Use

17

O

C B

Aa

c M

4a

O is the origin, = a, = c and = 4a. M is the midpoint of AB. (a) Find, in terms of a and c, in their simplest form

(i) the vector , Answer(a)(i) = [2]

(ii) the position vector of M. Answer(a)(ii) [2]

(b) Mark the point D on the diagram where = 3a + c. [2]

Answer(b) C = [3]

Question 19 is printed on the next page.

9

© UCLES 2012 0580/23/O/N/12 [Turn over

For

Examiner's

Use

20

O C

D

Ed

c

NOT TOSCALE

In the diagram, O is the origin.

= c and = d. E is on CD so that CE = 2ED. Find, in terms of c and d, in their simplest forms,

(a) , Answer(a) = [2]

(b) the position vector of E. Answer(b) [2]

15

0580/43/O/N/13© UCLES 2013 [Turn over

ForExaminer′s

Use

(b)

R

T

A

O B3b

4a

NOT TOSCALE

In the diagram, = 4a and = 3b.

R lies on AB such that = 5

1 (12a + 6b).

T is the point such that = 2

3 .

(i) Find the following in terms of a and b, giving each answer in its simplest form.

(a)

Answer(b)(i)(a) = ............................................... [1]

(b)

Answer(b)(i)(b) = ............................................... [2]

(c)

Answer(b)(i)(c) = ............................................... [1]

(ii) Complete the following statement.

The points O, R and T are in a straight line because ................................................................

........................................................................................................................................... [1]

(iii) Triangle OAR and triangle TBR are similar.

Find the value of area of trianglearea of triangle

OARTBR

.

Answer(b)(iii) ............................................... [2]_____________________________________________________________________________________

9

0580/41/O/N/13© UCLES 2013 [Turn over

ForExaminer′s

Use

(b)P Q

O S

Rp

s

NOT TOSCALE

In the pentagon OPQRS, OP is parallel to RQ and OS is parallel to PQ. PQ = 2OS and OP = 2RQ. O is the origin, = p and = s.

Find, in terms of p and s, in their simplest form,

(i) the position vector of Q,

Answer(b)(i) ............................................... [2]

(ii) .

Answer(b)(ii) = ............................................... [2]

(c) Explain what your answers in part (b) tell you about the lines OQ and SR.

Answer(c) .................................................................................................................................. [1]_____________________________________________________________________________________

10

© UCLES 2012 0580/42/O/N/12

For

Examiner's

Use

6 (a) Calculate the magnitude of the vector

− 5

3.

Answer(a) [2]

(b)

16

14

12

10

8

6

4

2

04 8 12 16 182 6 10 14

x

y

R P

(i) The points P and R are marked on the grid above.

=

− 5

3. Draw the vector on the grid above. [1]

(ii) Draw the image of vector after rotation by 90° anticlockwise about R. [2]

(c) = 2a + b and = 3b O a.

Find in terms of a and b. Write your answer in its simplest form. Answer(c) = [2]

11

© UCLES 2012 0580/42/O/N/12 [Turn over

For

Examiner's

Use

(d) =

−

5

2 and =

−1

5.

Write as a column vector.

Answer(d) =

[2]

(e)

A

B

CX

MNOT TOSCALE

= b and = c.

(i) Find in terms of b and c. Answer(e)(i) = [1]

(ii) X divides CB in the ratio 1 : 3 . M is the midpoint of AB.

Find in terms of b and c. Show all your working and write your answer in its simplest form. Answer(e)(ii) = [4]

10

© UCLES 2012 0580/42/O/N/12

For

Examiner's

Use

6 (a) Calculate the magnitude of the vector

− 5

3.

Answer(a) [2]

(b)

16

14

12

10

8

6

4

2

04 8 12 16 182 6 10 14

x

y

R P

(i) The points P and R are marked on the grid above.

=

− 5

3. Draw the vector on the grid above. [1]

(ii) Draw the image of vector after rotation by 90° anticlockwise about R. [2]

(c) = 2a + b and = 3b O a.

Find in terms of a and b. Write your answer in its simplest form. Answer(c) = [2]