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Higher Mathematics

GCC Vectors

1.[SQA] ABCD is a quadrilateral with vertices A(4,−1, 3) , B(8, 3,−1) , C(0, 4, 4) andD(−4, 0, 8) .(a) Find the coordinates of M, the midpoint of AB. 1

(b) Find the coordinates of the point T, which divides CM in the ratio 2 : 1. 3

(c) Show that B, T and D are collinear and find the ratio in which T divides BD. 4

Part Marks Level Calc. Content Answer U3 OC1

(a) 1 C CN G6, G25 1989 P2 Q2

(b) 3 C CN G25

(c) 4 C CN G23, G25

2.[SQA]


(a) 1 C CN G16 1990 P1 Q5

(b) 1 C CN G16

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Higher Mathematics

3.[SQA]

eplacements


(a) 1 C CN G16 1993 P1 Q1

(b) 2 C CN G16

4.[SQA]


3 A/B CN G16 1994 P1 Q18

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Higher Mathematics

5.[SQA]


(a) 2 C CN G16 1998 P1 Q3

(b) 1 C CN G26

(c) 1 C CN G16

6.[SQA]


(a) 2 C CN G16 1998 P1 Q5

(b) 2 C CN G16

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Higher Mathematics

7.[SQA]


(a) 3 C CR G16 1992 P2 Q2

(b) 2 C CR G16

(c) 3 C CR G27

(d) 5 C CR G28

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Higher Mathematics

8.[SQA]


(a) 3 C CR G16 1994 P2 Q3

(b) 1 C CR G25

(c) 4 C CR G28

(d) 2 C CR CGD

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Higher Mathematics

9.[SQA]


(a) 2 C CR G16 1996 P2 Q3

(b) 7 C CR G28

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Higher Mathematics

10.[SQA]


(a) 2 C CR G16 1998 P2 Q1

(b) 5 C CR G28

(c) 2 C CR CGD

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Higher Mathematics

11.[SQA] A box in the shape of a cuboidis designed with circles of differentsizes on each face.

The diagram shows three of thecircles, where the origin representsone of the corners of the cuboid. Thecentres of the circles are A(6, 0, 7) ,B(0, 5, 6) and C(4, 5, 0) .

Find the size of angle ABC. 7

O

x

y

z

A

B

C


5 C CR G17, G16, G22 2001 P2 Q4

2 A/B CR G26, G28 71·5◦

•1 ss: use−→BA.

−→BC

|−→BA||−→BC|•2 ic: state vector e.g. −→BA•3 ic: state a consistent vector e.g. −→BC•4 pd: process |−→BA|•5 pd: process |−→BC|•6 pd: process scalar product•7 pd: find angle

•1 use−→BA.

−→BC

|−→BA||−→BC|stated or implied by •7

•2 −→BA =

6−51

•3 −→BC =

40−6

•4 |−→BA| =√62

•5 |−→BC| =√52

•6 −→BA.

−→BC = 18

•7 AB̂C = 71·5◦

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Higher Mathematics

12.[SQA] The vectors p , q and r are defined as follows:

p = 3i − 3 j + 2k , q = 4i − j + k , r = 4i − 2 j + 3k .

(a) Find 2p − q + r in terms of i , j and k . 1

(b) Find the value of |2p − q + r | . 2


(a) 1 C CN G18 1989 P1 Q3

(b) 2 C CN G16

13.[SQA]


3 C CN G18 1999 P1 Q6

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Higher Mathematics

14.[SQA] The vector ai + b j + k is perpendicular to both the vectors i − j + k and−2i + j + k .Find the values of a and b . 3


3 C CN G18 a = 2, b = 3 1990 P1 Q12

15.[SQA] Calculate the length of the vector 2i − 3 j +√3k . 2


2 C CN G18 4 1995 P1 Q1

16.[SQA] Show that the vectors a = 2i + 3 j − k and b = 3i − j + 3k are perpendicular. 3


3 C CN G18, G27 a.b = · · · = 0 1991 P1 Q3

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Higher Mathematics

17.[SQA]eplacements


(a) 2 C CN G18, G16 1997 P1 Q4

(b) 1 C CN G16

18.[SQA]


(a) 3 C CN G18, G26 1993 P1 Q12

(b) 2 A/B CN G27

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Higher Mathematics

19.[SQA]


(a) 2 C CN G20 1999 P2 Q3

(b) 2 C CN G20

(c) 5 C CN G28

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Higher Mathematics

20.[SQA] If u =

−333

and v =

15−1

, write down the components of u + v and u − v .

Hence show that u + v and u − v are perpendicular. 3


3 C CN G20, G27 1994 P1 Q7

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Higher Mathematics

21.[SQA] A cuboid measuring 11 cm by 5 cm by 7 cm is placed centrally on top of anothercuboid measuring 17 cm by 9 cm by 8 cm.

Coordinates axes are taken as shown.

O

x

y

5 7

89

11

17

z

A

BC

(a) The point A has coordinates (0, 9, 8) and C has coordinates (17, 0, 8) .

Write down the coordinates of B. 1

(b) Calculate the size of angle ABC. 6


(a) 1 C CN G22 B(3, 2, 15) 2000 P2 Q9

(b) 6 C CR G28 92·5◦

•1 ic: interpret 3-d representation

•2 ss: know to use scalar product•3 pd: process vectors•4 pd: process vectors•5 pd: process lengths•6 pd: process scalar product•7 pd: evaluate scalar product

•1 B= (3, 2, 15) treat

3215

as bad form

•2 cosAB̂C =−→BA.

−→BC

|−→BA||−→BC|

•3 −→BA =

−37−7

•4 −→BC =

14−2−7

•5 |−→BA| =√107, |−→BC| =

√249

•6 −→BA.

−→BC = −7

•7 AB̂C = 92·5◦

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Higher Mathematics

22.[SQA] The diagram shows a square-basedpyramid of height 8 units.

Square OABC has a side length of 6 units.

The coordinates of A and D are (6, 0, 0)and (3, 3, 8) .

C lies on the y-axis.

(a) Write down the coordinates of B. 1

(b) Determine the components of−→DA

and−→DB. 2

(c) Calculate the size of angle ADB. 4

Ox

y

A(6, 0, 0)

BC

D(3, 3, 8)z


(a) 1 C CN G22 (6, 6, 0) 2002 P2 Q2

(b) 2 C CN G17−→DA =

3−3−8

,

−→DB =

33−8

(c) 4 C CR G28 38·7◦

•1 ic: interpret diagram

•2 ic: write down components of avector

•3 ic: write down components of avector

•4 ss: use e.g. scalar product formula•5 pd: process lengths•6 pd: process scalar product•7 pd: process angle

•1 B = (6, 6, 0)

•2 −→DA =

3−3−8

•3 −→DB =

33−8

•4 cosAD̂B =−→DA.

−→DB

|−→DA||−→DB|•5 |−→DA| =

√82, |−→DB| =

√82

•6 −→DA.

−→DB = 64

•7 AD̂B = 38·7◦

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Higher Mathematics

23. D,OABC is a square based pyramid as shown in the diagram below.

M

BC

A

D(2, 2, 6)z

Ox

y

O is the origin, D is the point (2, 2, 6) and OA = 4 units.

M is the mid-point of OA.

(a) State the coordinates of B. 1

(b) Express−→DB and

−→DM in component form. 3

(c) Find the size of angle BDM. 5


(a) 1 C CN G22 (4, 4, 0) 2011 P2 Q1

(b) 3 C CN G20, G22−→DB =

22−6

,−→DM =

0−2−6

(c) 5 C CN G28 40·3◦

•1 ic: state coordinates of B

•2 pd: state components of −→DB•3 ic: state coordinates of M•4 pd: state components of −→DM

•5 ss: know to use scalar product•6 pd: find scalar product•7 pd: find magnitude of a vector•8 pd: find magnitude of a vector•9 pd: evaluate angle BDM

•1 (4, 4, 0)

•2

22−6

•3 (2, 0, 0)

•4

0−2−6

•5 cos BDM =−→DB·−→DM|−→DB||−→DM|

•6 −→DB·−→DM = 32

•7 |−→DB| =√44

•8 |−→DM| =√40

•9 40·3◦ or 0·703 rads

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Higher Mathematics

24. The diagram shows a cuboid OPQR,STUV relative to the coordinate axes.

P is the point (4, 0, 0) , Q is (4, 2, 0)and U is (4, 2, 3) .

M is the midpoint of OR.

N is the point on UQ such thatUN = 1

3UQ.

z

S

M

T

N

R

P (4, 0, 0)

Q (4, 2, 0)

U (4, 2, 3)V

Ox

y

(a) State the coordinates of M and N. 2

(b) Express the vectors−→VM and

−→VN in component form. 2

(c) Calculate the size of angle MVN. 5


(a) 2 C CN G22, G25 M(0, 1, 0),N(4, 2, 2) 2010 P2 Q1

(b) 2 C CN G17−→VM =

0−1−3

,−→VN =

40−1

(c) 5 C CN G28 76·7◦ or 1·339 rad

•1 ic: interpret midpoint for M•2 ic: interpret ratio for N

•3 ic: interpret diagram•4 pd: process vectors

•5 ss: know to use scalar product•6 pd: find scalar product•7 pd: find magnitude of a vector•8 pd: find magnitude of a vector•9 pd: evaluate angle

•1 (0, 1, 0)•2 (4, 2, 2)

•3 −→VM =

0−1−3

•4 −→VN =

40−1

•5 cosMV̂N =

−→VM.

−→VN

|−→VM||−→VN|•6 −→VM.

−→VN = 3

•7 |−→VM| =√10

•8 |−→VN| =√17

•9 76·7◦ or 1·339 rads or 85·2 grads

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Higher Mathematics

25. (a)[SQA] Roadmakers look along the tops of a setof T-rods to ensure that straight sectionsof road are being created. Relative tosuitable axes the top left corners of theT-rods are the points A(−8,−10,−2) ,B(−2,−1, 1) and C(6, 11, 5) .

Determine whether or not the section ofroad ABChas been built in a straight line. 3

A

B

C

(b) A further T-rod is placed such that D hascoordinates (1,−4, 4) .Show that DB is perpendicular to AB. 3

A

B

C

D


(a) 3 C CN G23 the road ABC is straight 2001 P1 Q3

(b) 3 C CN G27, G17 proof

•1 ic: interpret vector (e.g. −→AB)•2 ic: interpret multiple of vector•3 ic: complete proof

•4 ic: interpret vector (i.e. −→BD)•5 ss: state requirement for perpend.•6 ic: complete proof

•1 e.g. −→AB =

693

•2 e.g. −→BC =

8124

= 43

−→AB or

−→AB = 3

231

and−→BC = 4

231

•3 a common direction exists and acommon point exists, so A, B, Ccollinear

•4 −→BD =

3−33

•5 −→AB.

−→BD = 0

•6 −→AB.

−→BD = 18− 27+ 9 = 0

or

•5 −→AB.

−→BD = 18− 27+ 9

•6 −→AB.

−→BD = 0 so AB is at right angles to

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Higher Mathematics

26.[SQA]


(a) 4 C CN G23 1991 P1 Q7

(b) 1 C CN G25

27.[SQA]


3 C CN G23 1994 P1 Q4

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Higher Mathematics

28.[SQA]


3 C CN G23 1997 P1 Q2

29.[SQA] Show that P(2, 2, 3) , Q(4, 4, 1) and R(5, 5, 0) are collinear and find the ratio inwhich Q divides PR. 4


4 C CN G23, G25−→QR = 1

2

−→PQ,

PQ : QR = 2 : 11990 P1 Q4

30.[SQA] A is the point (2,−5, 6) , B is (6,−3, 4) and C is (12, 0, 1) . Show that A, B and Care collinear and determine the ratio in which B divides AC. 4


4 C CN G23, G25 2 : 3 or equivalent 1996 P1 Q6

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Higher Mathematics

31.[SQA] D, E and F have coordinates (10,−8,−15) , (1,−2,−3) and (−2, 0, 1) respectively.(a) (i) Show that D, E and F are collinear.

(ii) Find the ratio in which E divides DF. 4

(b) G has coordinates (k, 1, 0) .

Given that DE is perpendicular to GE, find the value of k . 4


(a) 4 C CN G23, G24 3 : 1 2009 P1 Q22

(b) 4 C CN G27 k = 7

•1 ss: use vector approach•2 ic: compare two vectors•3 ic: complete proof•4 ic: state ratio

•5 ss: use vector approach•6 ss: know scalar product = 0 for ⊥vectors

•7 pd: start to solve•8 pd: complete

•1 −→DE =

−9612

or−→EF =

−324

•2 2nd column vector and−→( DE) = 3

−→EF

•3 −→DE and

−→EF have common point and

common direction; hence D, E and Fare collinear

•4 3 : 1

•5 −→GE =

1− k−3−3

•6 −→DE.

−→GE = 0

•7 −9(1− k) + 6× (−3) + 12× (−3)•8 k = 7

32.[SQA] The point Q divides the line joining P(−1,−1, 0) to R(5, 2,−3) in the ratio 2 : 1.Find the coordinates of Q. 3


3 C NC G25 (3, 1,−2) 2002 P1 Q2

•1 pd: find vector components•2 ss: use parallel vectors•3 pd: process vectors

•1 −→PR =

63−3

•2 −→PQ = 2

3

−→PR

•3 Q = (3, 1,−2)

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Higher Mathematics

33.[SQA]


3 C CN G25 1992 P1 Q15

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Higher Mathematics

34.[SQA]


(a) 3 C CR G25 1990 P2 Q4

(b) 5 C CR G28

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Higher Mathematics

35.[SQA]


(a) 3 C CR G25 1997 P2 Q2

(b) 7 C CR G28

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Higher Mathematics

36.[SQA] VABCD is a pyramid with a rectangular base ABCD.

Relative to some appropriate axes,

−→VA represents −7i − 13 j − 11k−→AB represents 6i + 6 j − 6k−→AD represents 8i − 4 j + 4k .

K divides BC in the ratio 1 : 3.

Find−→VK in component form. 3

A B

CD

V

K1

3


3 C CN G25, G21, G20

1−8−16

2000 P1 Q7

•1 ss: recognise crucial aspect•2 ic: interpret ratio•3 pd: process components

•1 −→VK =

−→VA +

−→AB +

−→BK or−→

VK =−→VB+

−→BK

•2 −→BK = 1

4

−→BC or 14

−→AD or

2−11

or

−1−7−17

•3 −→VK =

1−8−16

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Higher Mathematics

37.[SQA]


(a) 2 C CN G26 1999 P1 Q17

(b) 4 A/B CN G29, G30

38.[SQA]


1 C CN G26 1995 P1 Q16

3 A/B CN G29, G26

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Higher Mathematics

39.[SQA]


1 C CN G26 1997 P1 Q13

3 A/B CN G29, G27

40.[SQA] For what value of t are the vectors u =

t−23

and v =

210t

perpendicular? 2


2 C CN G27 t = 4 2000 P2 Q7

•1 ss: know to use scalar product•2 ic: interpret scalar product

•1 u.v = 2t− 20+ 3t•2 u.v = 0⇒ t = 4

41.[SQA] A(4, 4, 10) , B(−2,−4, 12) and C(−8, 0, 10) are the vertices of a right-angledtriangle.

Determine which angle of the triangle is the right angle. 3


3 C CN G27 1989 P1 Q6

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Higher Mathematics

42.[SQA] Find the value of k for which the vectors

12−1

and

−43k− 1

are perpendicular. 3


3 C CN G27 k = 3 1995 P1 Q4

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Higher Mathematics

43.[SQA]


(a) 7 C CR G28, G16 1991 P2 Q5

(b) 3 C CR CGD

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Higher Mathematics

44.[SQA]


3 C CN G29, G26 1992 P1 Q18

45.[SQA] Vectors p , q and r are representedon the diagram shown where angleADC = 30◦ .

It is also given that |p | = 4 and |q | = 3.

(a) Evaluate p .(q + r ) and r .(p − q) . 6

(b) Find |q + r | and |p − q | . 4

A

pD

30 °

B

r

q

C


(a) 6 B CN G29, G26 6√3, 94 2009 P2 Q7

(b) 2 A CR G21, G30 |q+ r| = 3√32

(b) 2 B CR G21, G30 |p− q| =√

(4− 3√32 )2 + ( 32 )

2

•1 ss: use distributive law•2 ic: interpret scalar product•3 pd: processing scalar product•4 ic: interpret perpendicularity•5 ic: interpret scalar product•6 pd: complete processing

•7 ic: interpret vectors on a 2-Ddiagram

•8 pd: evaluate magnitude of vectorsum

•9 ic: interpret vectors on a 2-Ddiagram

•10 pd: evaluate magnitude of vectordifference

•1 p.q+ p.r•2 4× 3 cos 30◦•3 6

√3(≈ 10·4)

•4 p.r = 0•5 −|r| × 3 cos 120◦•6 r = 3

2 and94

•7 q+ r ≡ from D to the proj. of A ontoDC

•8 |q+ r| = 3√32

•9 p− q =−→AC

•10 |p− q| =√

(4− 3√32 )2 + ( 32)

2(≈ 2·05)

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Higher Mathematics

46.[SQA]


1 C NC A6 1989 P1 Q9

4 A/B NC G29, G26

47.[SQA]


5 C CN G29, G27 1991 P1 Q17

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Higher Mathematics

48.[SQA]


3 C CN G30 1994 P1 Q3

49.[SQA] PQRS is a parallelogram with vertices P(1, 3, 3) , Q(4,−2,−2) and R(3, 1, 1) .

Find the coordinates of S. 3


3 C CN G30 1989 P1 Q4

[END OF QUESTIONS]

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