Vectors
Advanced Level Pure Mathematics
Advanced Level Pure Mathematics
7
Algebra
Chapter 7
Vectors
7.1
Fundamental Concepts
2
7.2
Addition and Subtraction of Vectors
2
7.3
Scalar Multiplication
3
7.4
Vectors in Three Dimensions
5
7.5
Linear Combination and Linear Independence
7
7.6
Products of Two Vectors
13
A. Scalar Product
B. Vector Product
7.7
Scalar Triple Product
22
Matrix Transformation*
24
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Page 1
Vectors
7.1
1.
2.
Advanced Level Pure Mathematics
Fundamental Concepts
Scalar quantities: mass, density, area, time, potential, temperature, speed, work, etc.
Vectors are physical quantities which have the property of directions and magnitude.
e.g. Velocity v , weight w , force f , etc.
3.
Properties:

(a) The magnitude of u is denoted by u .
(b)
AB  CD if and only if AB  CD , and AB and CD has the same direction.
(c)
AB  BA

(d) Null vector, zero vector 0 , is a vector with zero magnitude i.e. 0  0 .
The direction of a zero vector is indetermine.

(e) Unit vector, û or eu , is a vector with magnitude of 1 unit. I.e. u  1 .
(f)
7.2
1.
u
uˆ  
u
 
 u  u uˆ
Addition and Subtraction of Vectors
Geometric meaning of addition and subtraction.
AB  BC  CD  AD
PQ  q  p
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Vectors
2.
Advanced Level Pure Mathematics
Properties:
 

For any vectors u , v and w , we have
(a)
u v  vu,
(b) u  (v  w)  (u  v)  w ,
(c)
u0  0u
(d) u  (u)  (u)  u  0
(1) u  v  u  (v)
N.B.
(2) c  a  b  a  c  b
7.3
Scalar Multiplication
When a vector a is multiplied by a scalar m, the product ma is a vector parallel to a such that
(a) The magnitude of ma is m times that of a .
(b) When m  0 , ma has the same direction as that of a ,
When m  0 , ma has the opposite direction as that of a .
These properties are illustrated in Figure.
Theorem
Properties of Scalar Multiplication
Let m, n be two scalars. For any two vectors a and b , we have
(a) m(na)  (mn)a
(b) (m  n)a  ma  na
(c) m(a  b)  ma  mb
(d) 1a  a
(e) 0a  o
(f)  0  0
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Vectors
Theorem
Advanced Level Pure Mathematics
Section Formula
Let A,B and R be three collinear points.
If
Example
Solution
AR m
mOB  nOA
 , then OR 
.
RB n
mn
Prove that the diagonals of a parallelogram bisect each other.
Properties
(a) If a, b are two non-zero vectors, then a // b if and only if a  mb for some m  R .
(b)
a  b  a  b , and a  b  a  b
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Vectors
7.4
Advanced Level Pure Mathematics
Vectors in Three Dimensions
(a) We define i, j , k are vectors joining the origin O to the points (1,0,0) , (0,1,0) , (0,0,1) respectively.
(b) i, j and k are unit vectors. i.e. i  j  k  1 .
(c) To each point P(a, b, c) in R 3 , there corresponds uniquely a vector
OP  p  ai  bj  ck
where is called the position vector of P .
(d)
p  a2  b2  c2
(e)
p
ai  bj  ck
a2  b2  c2
(f) Properties
:
Let
(i)
(ii)
(iii)
N.B.
Example
p1  x1i  y1 j  z1k and p2  x2i  y2 j  z 2 k . Then
p1  p2 if and only if x1  x2 , y1  y2 and z1  z 2 ,
p1  p2  ( x1  x2 )i  ( y1  y2 ) j  ( z1  z 2 )k
p1   ( x1i  y1 j  z1k )  x1i  y1 j  z1k
For convenience, we write p  ( x, y, z )
Given two points A(6,8,10) and B(1,2,0) .
(a) Find the position vectors of A and . B .
(b) Find the unit vector in the direction of the position vector of A .
(c) If a point P divides the line segment AB in the ration 3 : 2 , find the coordinates of P .
Solution
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Vectors
Example
Advanced Level Pure Mathematics
Let A( 0 ,2 ,6 ) and B( 4 ,2 ,8 )
(a) Find the position vectors of A and B . Hence find the length of AB .
(b) If P is a point on AB such that AP  2PB, find the coordinates of P .
(c) Find the unit vector along OP .
Solution
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Vectors
7.5
Advanced Level Pure Mathematics
Linear Combination and Linear Independence
Definition
Consider a given set of vectors v1 , v2 ,, vn . A sum of the form
a1v1  a2 v2    a n vn
where a1 , a2 ,, an are scalars, is called a linear combination of v1 , v2 ,, vn .
If a vector v can be expressed as v  a1v1  a2 v2    an vn
Then v is a linear combination of v1 , v2 ,, vn . .
Example
r  u  2v  w is a linear combination of the vectors u, v , w .
Example
Consider u  (1,2,1), v  (6,4,2)  R 3 , show that w  (9,27) is a linear combination of u
and v while w1  (4,1,8) is not.
Solution
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Vectors
Definition
Advanced Level Pure Mathematics
If v1 , v 2 ,, v n are vectors in R n and if every vector in R n can be expressed as the linear
combination of v1 , v 2 ,, v n . Then we say that these vectors span (generate) R n or
v1 , v 2 ,, v n 
is the set of the basis vector.
Example
i, j
Example
(1,0,0), (0,1,0), (0,0,1)
Remark :
The basis vectors have an important property of linear independent which is defined as follow:
Definition
Definition
Example
is the set of basis vectors in R 2 .
is the set basis vector in R 3 .
The set of vector v1 , v 2 ,, v n  is said to be linear independent if and only if the vectors
equation k1v1  k 2 v 2    k n v n  0 has only solution k1  k 2    k n  0
The set of vector v1 , v 2 ,, v n  is said to be linear dependent if and only if the vectors
equation k1v1  k 2 v 2    k n v n  0 has non-trivial solution.
(i.e. there exists some k i such that k i  0 )
Determine whether v1  (1,2,3), v 2  (5,6,1), v 3  (3,2,1) are linear independent or dependent.
Solution
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Vectors
Example
Advanced Level Pure Mathematics
Let a  i  j  k , b  2i  j  k and c  j  k . Prove that
(a) a, b and c are linearly independent.
(b) any vector v in R 3 can be expressed as a linear combination of a, b and c .
Solution
Example
If vectors a, b and c are linearly independent, show that a  b, b  c and c  a are also
linearly independent.
Solution
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Page 9
Vectors
Example
Advanced Level Pure Mathematics
Let a  ( 2 ,3  t ,1 ) , b  ( 1  t ,2 ,3 ) and c  ( 0 ,4 ,2  t ).
(a) Show that b and c are linearly independent for all real values of t .
(b) Show that there is only one real number t so that a , b and c are linearly dependent.
For this value of t , express a as a linear combination of b and c .
Solution
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Vectors
Advanced Level Pure Mathematics
Theorem
(1) A set of vectors including the zero vector must be linearly dependent.
(2) If the vector v can be expressed as a linear combination of v1 , v2 ,vn , then the set of vectors
v1 , v2 ,vn and v are linearly dependent.
(3) If the vectors v1 , v2 ,vn are linearly dependent, then one of the vectors can expressed as a linear
combination of the other vectors.
Proof
Example
Let a  i  3 j  5k , b  i and c  3 j  5k .
Prove that a, b and c are linearly dependent.
Solution
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Vectors
Advanced Level Pure Mathematics
Theorem
Proof
Two non-zero vectors are linearly dependent if and only if they are parallel.
Theorem
Proof
Three non-zero vectors are linearly dependent if and only if they are coplanar.
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Vectors
Advanced Level Pure Mathematics
7.6
Products of Two Vectors
A.
Scalar Product
Definition
The scalar product or dot product or inner product of two vectors a and b , denoted by
a  b  a b cos
a  b , is defined as
(0     )
where  is the angle between a and b .
a b
.
ab
Remarks
By definition of dot product, we can find  by cos 
Example
If a  3, b  4 and angle between a and b is 60 , then
a b 
Theorem
Properties of Scalar Product
Let a, b, c be three vectors and m be a scalar. Then we have
(1) a  a  a
2
(2) a  b  b  a
(3) a  (b  c)  a  b  a  c
(4) m(a  b)  (ma)  b  a  (mb)
(5) a  a  0 if a  0 and a  a  0 if a  0
Theorem
If p  a1i  b1 j  c1k and q  a2i  b2 j  c2 k . Then
(1) p  q  a1a2  b1b2  c1c2
pq
( p, q  0)
(2) cos
=
pq
=
(3)
a1 a 2  b1b2  c1c2
a1  b1  c1
2
2
a 2  b2  c2
2
2
2
2
p  q  0 if and only if p  q .
(4) a1a2  b1b2  c1c2  0 if and only if p  q .
Example
Find the angle between the two vectors a  2i  2 j  k and b  2i  2k.
Solution
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Vectors
Remarks
Advanced Level Pure Mathematics
Two non-zero vectors are said to be orthogonal if their scalar product is zero. Obviously, two
perpendicular vectors must be orthogonal since  

2
, cos  0 , and so their scalar product
is zero. For example, as i, j and k are mutually perpendicular, we have
i  j  j k  k i  0.
Also, as i, j and k are unit vectors, i  i  j  j  k  k  1 .
Example
State whether the two vectors i  3 j  4k and  i  j  k are orthogonal.
Solution
Example
Given two points A  (2s, s  1, s  3) and B  ( t  2 ,3t  1,t )
and two vectors
r1  2i  2 j  k and r2  i  j  2k
If AB is perpendicular to both r1 and r2 , find the values of s and t .
Solution
Example
Let a, b and c be three coplanar vectors. If a and b are orthogonal, show that
c
ca
c b
a
b
aa
bb
Solution
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Vectors
Advanced Level Pure Mathematics
Example
Determine whether the following sets of vectors are orthogonal or not.
(a) a  4i  2 j and b  2i  3 j
(b) a  5i  2 j  4k and b  i  2 j  k
(c) a  3i  j  4k and b  2i  2 j  2k
Solution
B.
Vector Product
Definition
If u  (u1 , u 2 , u3 ) and v  (v1 , v2 , v3 ) are vectors in R 3 , then the vector product and cross
product u  v is the vector defined by
uv
=
(u 2 v3  u3 v2 , u3 v1  u1v3 , u1v2  u 2 v1 )
i
=
Example
j
k
u1 u 2
v1 v 2
u3
v3
Find a  b , a  (a  b) and b  (a  b) if a  3i  2 j  k and b  i  4 j  k .
Solution
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Vectors
Example
Advanced Level Pure Mathematics
Let a  i  k , b  2i  j  k and c  i  2 j  2k . Find
(a)
(e)
(g)
(i)
(k)
ab
( a  b)c
(b) b  c
(f) a  (b  c)
ab  bc  c a
[( a  b)  c]  a
a  (b  c)
(c)
ab
(d) a  c
(h) (a  b)  c  (c  b)  a
(j) [( a  b)  (c  a)]  b
(l) (a  b)  c
Solution
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Vectors
Theorem
Advanced Level Pure Mathematics
If u and v are vectors, then
(a) u  (u  v)  0
(b) v  (u  v)  0
(c)
2
2
2
u  v  u v  (u  v) 2
Proof
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Vectors
Remarks
Advanced Level Pure Mathematics
(i)
By (c)
uv
uv

2
2
2
=
u v  (u  v) 2
=
u v  u v cos 2  , where  is angle between u and v .
=
u v (1  cos 2  )
=
u v sin 2 
=
u v sin 
2
2
2
2
2
2
2
2
The another definition of u  v is u  v  u v sin  en where en is a unit vector
perpendicular to the plane containing u and v .
(ii) u  v  v  u and u  v  v  u
(iii) i  j 
Definition
jk 
k j 
The vector product (cross product) of two vectors a and b , denoted by a  b , is a vector
such that (1) its magnitude is equal to a b sin  , where  is angle between a and b.
(2) perpendicular to both a and b and a, b, a  b form a right-hand system.
If a unit vector in the direction of a  b is denoted by en , then we have
a  b  a b sin  en
(0     )
Geometrical Interpretation of Vector Product
(1) a  b is a vector perpendicular to the plane containing a and b .
(2) The magnitude of the vector product of a and b is equal to the area of parallelogram with a and b
as its adjacent sides.
Corollary
(a) Two non-zero vectors are parallel if and only if their vector product is zero.
(b) Two non-zero vectors are linearly dependent if and only if their vector product is zero.
Theorem
Properties of Vector Product
(1) a  (b  c)  a  b  a  c
(2) m(a  b)  (ma)  b  a  (mb)
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Vectors
Example
Advanced Level Pure Mathematics
Find a vector perpendicular to the plane containing the points A(1,2,3), B(1,4,8) and
C (5,1,2) .
Solution
Example
If a  b  c  0, show that a  b  b  c  c  a
Solution
Example
Find the area of the triangle formed by taking A(0,2,1), B(1,1,2) and C (1,1,0) as vertices.
Solution
Example
Let OA  i  2 j  k , OB  3i  j  2k and OC  5i  j  3k .
(a) Find AB  AC .
(b) Find the area of ABC.
Hence, or otherwise, find the distance from C to AB .
Solution
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Vectors
Example
Advanced Level Pure Mathematics
Let a and b be two vectors in R 3 such that
a  a  b  b  1 and a  b  0


Let S  a  b  R 3 :  ,   R .
(a) Show that for all u  S ,
u  (u  a)a  (u  b)b
(b) For any v  R 3 , let w  (v  a)a  (v  b)b. Show that for all u  S , (v  w)  u  0 .
Solution
Example
Let a, b, c  R 3 .
If a  (b  c)  (a  b)  c  0 , prove that a  b  b  c  c  a  0 .
Solution
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Vectors
Example
Advanced Level Pure Mathematics
Let u, v and w be linearly independent vectors in R 3 .
Show that :
(a) If u  (u1 , u 2 , u3 ) , v  (v1 , v2 , v3 ) and w  ( w1 , w2 , w3 ) ,
u1
then u 2
u3
v1
v2
v3
w1
w2  0
w3
(b) If s  R 3 such that s  u  s  v  s  w  0 , then s  0 .
(c) If u  (v  w)  (u  v)  w  0 , then u  v  v  w  w  u  0 .
(d) If u  v  v  w  w  u  0 ,
then r 
r u
r v
rw
u
v
w for all r  R 3 .
u u
vv
w w
Solution
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Vectors
7.7
Advanced Level Pure Mathematics
Scalar Triple Product
Definition
The scalar triple product of 3 vectors a, b and c is defined to be (a  b)  c .
Let the angle between a and b be  and that between a  b and c be  .
As shown in Figure, when 0   
Volume of Parallelepiped

2
, we have
=
Base Area  Height
=

( a  b)  c
=
=
=
=
Geometrical Interpretation of Scalar Triple Product
The absolute value of the scalar triple product (a  b)  c is equal to the volume of the parallelepiped with
a, b and c as its adjacent sides.
Theorem
Remarks
Properties of Vector Product
Let a , b and c be three vectors. Then
(a  b)  c  (b  c)  a  (c  a)  b
Volume of Parallelepiped
=
a1
a2
a3
b1
c1
b2
c2
b3
c3
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Vectors
Example
Advanced Level Pure Mathematics
Let A(3,5,6), B (2,3,2) , C (1,8,8)
(a) Find the volume of parallelepiped with sides OA, OB and OC .
(b) What is the geometrical relationship about point O, A, B, C in (a).
Solution
Example
A, B, C are the points (1,1,0) , (2,1,1), (1,1,1) respectively and O is the origin.
Let
a  OA, b  OB and c  OC .
(a) Show that a, b and c are linearly independent.
(b) Find
(i) the area of OAB , and
(ii) the volume of tetrahedron OABC .
Solution
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Vectors
Advanced Level Pure Mathematics
Matrix Transformation*
Linear transformation of a plane (reflections, rotation)
Consider the case with the point P( x, y )  P' ( x' , y ' ) such that x  x' , y  y '
 x' 
 
 y'
=
 1 0  x 

 
 0  1 y 
r'
=
Ar ,
1 0 

where A  
 0  1
A is a matrix of transformation of reflection.
In general, any column vector pre-multiplied by a 2  2 matrix, it is transformed or mapped ( x' , y ' ) into
another column vector.
Example
 a b   x'   a b  x 
 
 ,    
A  
 c d   y '   c d  y 
We have x'  ax  by
y '  cx  dy
If using the base vector in R 2 , i.e (1,0), (0,1) .
 a b  1   a 
 a b  0   b 
     ,

    
 
 c d  0   c 
 c d  1   d 
then a, b, c, d can be found.
The images of the points (1,0), (0,1) under a certain transformation are known. Therefore, the
matrix is known.
Eight Simple Transformation
I.
Reflection in x-axis
II.
Reflection in y-axis
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Vectors
Advanced Level Pure Mathematics
III.
Reflection in x  y .
IV.
Reflection in the line y   x
V.
Quarter turn about the origin
VI.
Half turn about the origin
VII.
Three quarter turn about the origin
VIII.
Identity Transformation
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Vectors
Advanced Level Pure Mathematics
Some Special Linear Transformations on R2
I.
Enlargement
If OP  r , then OP '  kr .
k 0

A  
0 k 
II.
(a) Shearing Parallel to the x-axis
The y-coordinate of a point is unchanged but the x-coordinate is changed by adding to it a
quantity which is equal to a multiple of the value of its y-coordinate.
(b) Shearing Parallel to the y-axis
III.
Rotation
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Vectors
IV.
Example
Advanced Level Pure Mathematics
Reflection about the line y  (tan  ) x
If the point P ( 4,2) is rotated clockwise about the origin through an angle 60 , find its final
position
Solution
Example
 x
A translation on R 2 which transforms every point P whose position vector is p   
 y
 x' 
To another point Q with position vector q    defined by
 y' 
 x'   x   2 
       
 y'   y   3 
Find the image of (a) the point (4,2)
(b) the line 2 x  y  0
Solution
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Vectors
Advanced Level Pure Mathematics
Linear Transformation
Definition
Let V and U be two sets. A mapping  : V  U is called a linear transformation from
V to U if and only if it satisfies the condition:
 (au  bv)  a (u )  b (v), u, v  V and a, b  R.
Example
Let V be the set of 3 1 matrices and A be any real 3 3 matrix. A mapping f : V  V
Such that f ( x)  Ax, x  V . Show that f is linear.
Solution
In R 3 , consider a linear transformation  : R 3  R 3 , let v  R 3 , v  (a, b, c)  ai  bj  ck .
We are going to find the image of v under  .
 (v)   (ai  bj  ck )  a (i )  b ( j )  c (k )
Therefore,  (v ) can be found if  (i ),  ( j ) and  (k ) are known. That is to say, to specify 
completely, it is only necessary to define  (i ),  ( j ) and  (k ) .
For instance, we define a linear transformation
 : R3  R3

by  (i)  2i  j  3k ,  ( j )  i  2k ,  (k )  3i  2 j  2k .
 (3i  2 j  4k )
=
=
=
We form a matrix A such that
Consider
 3 
 
A 2  =
  4
 
A
 4i  5 j  13k
=
 (i)  ( j)  (k )
=
 2 1 3 


 1 0  2
 3 2 2 


 2 1 3  3 

 
  1 0  2  2 
  3 2 2   4 

 
=
 4


 5 
  13 


The result obtained is just the same as  (3i  2 j  4k ) .
The matrix A representing the linear transformation  is called the matrix representation of the linear
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Vectors
Advanced Level Pure Mathematics
transformation 
Example
Let  : R 3  R 2 , defined by  (i )  i  2 j ,  ( j )   j ,  (k )  4i  3 j.
4 
1 0
 .
The matrix represent representation of a linear transformation is 
 2  1  3  23
Example
1 2 


The matrix B   0  1 represents a linear transformation
1 1 


 : R 2  R 3 , defined by  (i)  i  k ,  ( j )  2i  j  k .
Example
Let  , : R 3  R 3 be two linear transformations whose matrix representations are
respectively
 1 0  1
0  2 1 




A   0 1 2  and B   1 1
0 .
1 1 0 
 2 1  1




Find the matrix representation of    .
Solution
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Vectors
Example
Advanced Level Pure Mathematics
 x'   a b  x 
  for any ( x, y)  R 2 , then
If    
 y '   c d  y 
a b 

 is said to be the matrix
c d 
representation of the transformation which transforms ( x, y ) to ( x' , y ' ) .
Find the matrix representation of
(a) the transformation which transforms any point ( x, y ) to ( x, y ) ,
(b) the transformation which transforms any point ( x, y ) to ( y , x )
Solution
Example
It is given that the matrix representing the reflection in the line y  (tan  ) x is
 cos 2

 sin 2
sin 2 

 cos 2 
Let T be the reflection in the line y 
1
x.
2
(a) Find the matrix representation of T .
(b) The point (4,7) is transformed by T to another point ( x1 , y1 ) . Find x1 , y1 .
(c) The point ( 4,10) is reflected in the line y 
1
x  3 to another point ( x2 , y2 ) .
2
Find x 2 and y 2 .
Solution
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