Higher Maths
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Vectors
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The questions are in groups
General vector questions (15)
Points dividing lines in ratios
Collinear points (8)
Angles between vectors (5)
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Higher
General Vector Questions
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Vectors
Higher
Vectors u and v are defined by u  3i  2 j and v  2i  3 j  4k
Determine whether or not u and v are perpendicular to each other.
Is Scalar product = 0
 3
 
u.v   2 
 0
 
u.v  3  2  2  3  0  4
u.v  0
2
 
 3 
4
 
 u.v  6   6   0
Hence vectors are perpendicular
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Vectors
Higher
For what value of t are the vectors u
Put Scalar product = 0
 t 
 
  2 
 3
 
 t 
 
u.v   2 
3
 
u.v  2t   2 10  3t
Perpendicular  u.v = 0
and v
2
 10 
 t 
 
perpendicular ?
2
 
10 
t 
 
 u.v  5t  20
 0  5t  20
 t4
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Vectors
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VABCD is a pyramid with rectangular base ABCD.
The vectors AB, AD and AV are given by
AB  8i  2 j  2k
AD  2i  10 j  2k
AV  i  7 j  7k
Express CV
in component form.
Ttriangle rule  ACV
AC  CV  AV
Triangle rule  ABC
AB  BC  AC also

 CV  AV  AB  AD

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
Re-arrange
 CV  AV  AC
BC  AD
1 8  2 
     
 CV   7    2    10 
 7   2   2 
     
 9 
 
 CV   5 
7
 
CV  9i  5 j  7k
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Vectors
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The diagram shows two vectors a and b, with | a | = 3 and | b | = 22.
These vectors are inclined at an angle of 45° to each other.
a) Evaluate
i) a.a
ii) b.b
iii) a.b
b) Another vector p is defined by p  2a  3b
Evaluate p.p and hence write down | p |.
i)
a  a  a a cos 0  3  3 1  9
iii) a  b  a b cos 45  3  2 2 
b)
p  p   2a  3b    2a  3b 
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8
1
6
2
 4a.a  12a.b  9b.b
 36  72  72  180 Since p.p = p2
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bb  2 2 2 2
ii)
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p  180  6 5
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Vectors
Higher
Vectors p, q and r are defined by p  i  j - k ,
a) Express p  q  2r in component form
b)
Calculate p.r
c)
Find |r|
q  i  4k ,
  i  j - k    i  4k   2  4i  3 j 
and
r  4i  3 j
 8i  5 j - 5k
a)
p  q  2r
b)
p.r   i  j - k  .  4i  3 j   p.r  1 4  1 (3)  (1)  0  p.r  1
c)
r 
42  (3)2
 r  16  9 
r 5
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The diagram shows a point P with co-ordinates
(4, 2, 6) and two points S and T which lie on the x-axis.
If P is 7 units from S and 7 units from T,
find the co-ordinates of S and T.
Use distance formula
S ( a, 0, 0)
PS 2  49  (4  a)2  22  62
 a  43
T (b, 0, 0)
 49  (4  a)2  40
 9  (4  a)2
 a  7 or a  1
hence there are 2 points on the x axis that are 7 units from P
S (1, 0, 0)
and
i.e. S and T
T (7, 0, 0)
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Vectors
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The position vectors of the points P and Q are
p = –i +3j+4k and q = 7 i – j + 5 k respectively.
a) Express PQ in component form.
b) Find the length of PQ.
a)
b)
PQ  q - p
 PQ
 7   1
  1 -  3 
5 4
   
PQ  82  (4) 2  12
 PQ
 64  16  1
8
  4 
1
 
 81
 8i  4 j  k
9
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Vectors
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P
PQR is an equilateral triangle of side 2 units.
PQ  a, PR  b, and QR  c
Evaluate a.(b + c) and hence identify
two vectors which are perpendicular.
a
b
Diagram
Q
a.(b  c )  a.b  a.c
a.b  a b cos 60
60°
60°
60°
R
c
 a.b  2  2 
1
2
 a.b  2
NB for a.c vectors must point OUT of the vertex ( so angle is 120° )
 1

a.c

2

2

a.c  a c cos120
 
 2
Hence
a.(b  c )  0
 a.c   2
so, a is perpendicular to b + c
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Vectors
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Calculate the length of the vector 2i – 3j + 3k
Length  2  (3) 
2
2
 3
2
 493
 16
4
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Vectors
Higher
Find the value of k for which the vectors
1
 
 2
 1 
 
1
 
0  2 
 1
 
 4 


3


 k 1


Put Scalar product = 0
0  4  6  (k  1)
and
 4 


 3 
 k 1 


are perpendicular
 0  2  k 1
 k 3
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A is the point (2, –1, 4), B is (7, 1, 3) and C is (–6, 4, 2).
If ABCD is a parallelogram, find the co-ordinates of D.
AD  BC  c  b
D is the displacement
hence
d
BC
AD
 2   13 
  

  1   3 
 4   1 
  

 6   7 
   
  4   1
 2   3
   
BC
 13 


 3 
 1 


from A
d
 11
  2 
 3 


 D  11, 2, 3
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Vectors
If
Higher
 3 
u   3 
3
 
and v
1
 
5
 1
 
write down the components of u + v and u – v
Hence show that u + v and u – v are perpendicular.
uv
 2 
  8 
2
 
 u  v  . u  v 
uv
 2   4 
   
  8    2 
2 4
   
 4 
  2 
4
 
look at scalar product
 u  v  . u  v 
 (2)  (4)  8  (2)  2  4
 8 16  8
0
Hence vectors are perpendicular
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The vectors a, b and c are defined as follows:
a = 2i – k, b = i + 2j + k, c = –j + k
a) Evaluate a.b + a.c
b) From your answer to part (a), make a deduction about the vector b + c
a)
 2  1
   
a.b   0    2 
 1  1 
   
a.c
b)
2 0
   
  0    1
 1  1 
   
a.b  2  0  1
a.b  1
a.c  0  0  1
a.c  1
a.b  a.c  0
b + c is perpendicular to a
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A is the point ( –3, 2, 4 ) and B is ( –1, 3, 2 )
Find:
a) the components of AB
b) the length of AB
a)
b)
AB  b  a
AB
 1  3 
   
  3  2 
2 4
   
AB  22  12  (2)2
AB
 2
 
 1 
 2 
 
AB  4  1  4
AB  3
AB  9
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In the square based pyramid,
all the eight edges are of length 3 units.
AV  p, AD  q, AB  r ,
Evaluate p.(q + r)
Triangular faces are all equilateral
p.(q  r )  p.q  p.r
p.q  p q cos 60
p.r  p r cos60
1
2
p.(q  r )  4  4
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p.q
1
 3 3
2
p.r
1
 3 3
2
p.q  4
p.q 
1
2
1
4
2
p.(q  r )  9
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Points dividing lines in ratios
Collinear Points
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A and B are the points (-1, -3, 2)
and (2, -1, 1) respectively.
B and C are the points of trisection of AD.
That is, AB = BC = CD.
Find the coordinates of D
AB 1

AD 3
 3AB  AD
 3b  3a  d  a
 d
2
 1
 
 
 3  1  2  3 
1
2
 
 
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 3 b  a   d  a
 d  3b  2a
 d
8
 
 3
 1
 
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 D(8, 3, 1)
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Vectors
Higher
The point Q divides the line joining P(–1, –1, 0) to R(5, 2 –3) in the ratio 2:1.
Find the co-ordinates of Q.
R
Diagram
PQ 2

QR 1
 3q
P
1
 q  p  2r  2q
 PQ  2QR
 5   1
   
 2  2    1
 3   0 
   
2
Q
 3q  2r  p




1 9 
 q 3
3  6 
 Q(3, 1,  2)
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a) Roadmakers look along the tops of a set of T-rods to ensure
that straight sections of road are being created.
Relative to suitable axes the top left corners of the T-rods are
the points A(–8, –10, –2), B(–2, –1, 1) and C(6, 11, 5).
Determine whether or not the section of road ABC has been
built in a straight line.
b) A further T-rod is placed such that D has co-ordinates (1, –4, 4).
Show that DB is perpendicular to AB.
a)
AB  b  a
AB
b)
and
AB
 6
 2
  9   3  3 
 3
1
 
 
AC
 14 
 2
 
 
  21  7  3 
7
1
 
 
AC are scalar multiples, so are parallel. A is common. A, B, C are collinear
Use scalar product
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AB.BD
 6  3 
   
  9  .  3 
 3  3 
   
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AB.BD  18  27  9  0
Hence, DB is perpendicular to AB
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Vectors
Higher
VABCD is a pyramid with rectangular base ABCD.
Relative to some appropriate axis,
VA represents – 7i – 13j – 11k
6i + 6j – 6k
AB represents
8i – 4j – 4k
AD represents
K divides BC in the ratio 1:3
Find VK in component form.
VA  AB  VB
VK  VB  KB
 VK
VK  KB  VB
1
4
1
4
1
4
 VK  VA  AB  AD
 7   6 
8

   1 
  13    6    4 
 11  6  4  4 

  
 
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4
KB  CB  DA   AD
 VK
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 1 


  8 
 18 


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The line AB is divided into 3 equal parts by
the points C and D, as shown.
A and B have co-ordinates (3, –1, 2) and (9, 2, –4).
a) Find the components of AB and AC
b) Find the co-ordinates of C and D.
a)
b)
AB  b  a
AB
6
  3 
 6 
 
C is a displacement of AC from A
similarly
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d
 5  2 
  0    1 
 0   2 
   




2
1
 
AC  AB   1 
3
 2 
c
3 2
   
  1   1 
 2   2 
   
 C (5, 0, 0)
 D(7, 1,  2)
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Relative to a suitable set of axes, the tops of three chimneys have
co-ordinates given by A(1, 3, 2), B(2, –1, 4) and C(4, –9, 8).
Show that A, B and C are collinear
AB  b  a
AB
and
AB
1
  4 
 2
 
AC
 3 
1


 
  12   3  4 
 6 
2


 
AC are scalar multiples, so are parallel. A is common. A, B, C are collinear
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A is the point (2, –5, 6), B is (6, –3, 4) and C is (12, 0, 1).
Show that A, B and C are collinear
and determine the ratio in which B divides AC
AB  b  a
AB
and
BC
AB 2

BC 3
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AB
4
2
 
 
  2   2 1 
 2 
 1
 
 
BC
6
2
 
 
  3   3 1 
 3 
 1
 
 
are scalar multiples, so are parallel. B is common. A, B, C are collinear
A
2
B
3
C
B divides AB in ratio 2 : 3
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Relative to the top of a hill, three gliders
have positions given by
R(–1, –8, –2),
S(2, –5, 4) and T(3, –4, 6).
Prove that R, S and T are collinear
RS  s  r
RS
and
RT
RS
 3
1
 
 
  3  3  1 
 6
 2
 
 
RT
 4
1
  4   4  1 
8
 2
 
 
are scalar multiples, so are parallel. R is common. R, S, T are collinear
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Vectors
Higher
The diagram shows vectors a and b.
If |a| = 5, |b| = 4 and a.(a + b) = 36
Find the size of the acute angle
between a and b.
cos  
a.b
a b
a.a  a a  25
11
cos  
5 4
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a.(a  b)  36  a.a  a.b  36
 25  a.b  36
 11 
  cos  
 20 
  56.6
1
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 a.b  11
Hint
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Vectors
Higher
The diagram shows a square based pyramid of height 8 units.
Square OABC has a side length of 6 units.
The co-ordinates of A and D are (6, 0, 0) and (3, 3, 8).
C lies on the y-axis.
a) Write down the co-ordinates of B
b) Determine the components of DA and DB
c) Calculate the size of angle ADB.
a)
c)
B(6, 6, 0)
cos  
DA.DB
DA DB
cos  
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b)
3
DA   3 
 8 
 
64
82 82
DA.DB
DB
3
 
 3
 8 
 
3 3
   
  3  .  3   64
 8   8 
   
  38.7
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Vectors
Higher
A box in the shape of a cuboid designed with circles of different
sizes on each face.
The diagram shows three of the circles, where the origin represents
one of the corners of the cuboid.
The centres of the circles are A(6, 0, 7), B(0, 5, 6) and C(4, 5, 0)
Find the size of angle ABC
Vectors to point
away from vertex
6
 
BA   5 
1
 
 4
 
BC   0 
 6 
 
BA  36  25  1  62
cos  
18
62 52
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BA.BC  24  0  6  18
BC  16  36  52
  71.5
Hint
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Vectors
Higher
A cuboid measuring 11cm by 5 cm by 7 cm is placed centrally
on top of another cuboid measuring 17 cm by 9 cm by 8 cm.
Co-ordinate axes are taken as shown.
a) The point A has co-ordinates (0, 9, 8) and C has
co-ordinates (17, 0, 8).
Write down the co-ordinates of B
b) Calculate the size of angle ABC.
a)
B(3, 2, 15)
b)
 15 
 
BC   2 
 7 
 
BA.BC  45  14  49   10
BA  9  49  49  107
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 3 
 
BA   7 
 7 
 
BC  225  4  49  278
10
cos  
278 107
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  93.3
Hint
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Vectors
Higher
A triangle ABC has vertices A(2, –1, 3), B(3, 6, 5) and C(6, 6, –2).
a) Find
and
AB
AC
b) Calculate the size of angle BAC.
c) Hence find the area of the triangle.
a)
b)
1
 
AB  b  a   7 
 2
 
AB  12  7 2  22  54
cos  
c)
4
 
AC  c  a   7 
 5 
 
43
 0.6168
54 90
Area of  ABC =
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AC  90
AB. AC  4  49  10  43
  cos1 0.6168  51.9
1
ab sin C
2
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
1
90 54 sin 51.9
2
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 BAC = 51.9

27.43
unit
2
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Vectors
Higher
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Vectors
Higher
Table of exact values
sin
cos
tan
Previous
30°
45°
60°

6
1
2

4

3
1
2
1
2
3
2
3
2
1
3
1
1
2
3