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A2-Level Maths:
Core 4
for Edexcel
C4.7 Vectors 1
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Vectors in two and three dimensions
Contents
Vectors in two and three dimensions
The magnitude of a vector
Multiplying vectors by scalars
Adding and subtracting vectors
Position vectors and coordinate geometry
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Vectors and scalars
A vector is a quantity that has both size (or magnitude) and
direction.
Examples of vector quantities include:
displacement
velocity
force
A scalar is a quantity that has size (or magnitude) only.
Examples of scalar quantities include:
length
speed
mass
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Representing vectors
A vector can be represented using a line segment with an
arrow on it.
For example, the vector that goes from the point A to the point
B can be represented by the following directed line segment.
B
A
The magnitude of the vector is given by the length of the line.
The direction of the vector is given by the arrow on the line.
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Representing vectors
We can write this vector as AB.
Vectors can also be written using single letters in bold type.
For example, we can call this vector a.
When this is hand-written, the a is written as
a
To go from the point A to the point B we must move 6 units to
the right and 3 units up.
B
3
A
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Representing vectors
We can represent this movement using a column vector.
6
AB =  
3
This component tells us the number
of units moved in the x-direction.
This component tells us the number
of units moved in the y-direction.
We can also represent vectors in three dimensions relative to
a three dimensional coordinate grid:
A third axis, the z-axis, is added
at right angles to the xy-plane.
Conventionally, we show the zaxis pointing vertically upwards
with the xy-plane horizontal.
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Representing vectors
For example, consider the following three-dimensional vector
CD.
To go from the point C to the point D
5
C
we must move
z
y
x
–3
–2
5 units in the x-direction,
D
–2 units in the z-direction.
–3 units in the y-direction
This three-dimensional vector can be written in column vector
form as:
5
CD =  3 
 2 
 
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This component tells us the number
of units moved in the x-direction.
This component tells us the number
of units moved in the y-direction.
This component tells us the number
of units moved in the z-direction.
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Equal vectors
Two vectors are equal if they have the same magnitude and
direction.
For example, in the following diagram:
B
C
4
AB = DC =   = 4i + 3 j
3
and
A
D
5
BC = AD =   = 5i  j
 1
General displacement vectors that are not fixed to any point
are often called free vectors.
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The negative of a vector
5
Here is the vector AB = a =  
 2 
A
Suppose the arrow went in the
opposite direction, from B to A:
A
a
B
B
This is the negative (or inverse) of the vector AB.
We can describe this new vector as:
 AB
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BA
–a
or
 5 
 2
 
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The negative of a vector
In general,
 a1 
 a1 
If AB = a =   then BA =  a = 

a

a
 2
 2
And in three-dimensions,
 a1 
 a1 
If AB = a =  a2  then BA =  a =  a2 
a 
 a 
 3
 3
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The zero and unit vector
A vector with a magnitude of 0 is called the zero vector.
The zero vector is written as 0 or hand-written as
0
A vector with a magnitude of 1 is called a unit vector.
The most important unit vectors are those that run parallel to
the x- and y-axes. These are called unit base vectors.
 1
The horizontal unit base vector,   , is called i.
0
0
The vertical unit base vector,   , is called j.
 1
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The unit base vectors
The unit base vectors, i and j, run parallel to the x- and y-axes.
y-axis
j
x-axis
i
Any column vector can easily be written in terms of i and j.
For example,
5
 4  = 5i  4 j
 
The number of i’s tells us how many units are moved
horizontally, and the number of j’s tells us how many units are
moved vertically.
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The unit base vectors
In three dimensions, we introduce a third unit base vector, k,
that runs parallel to the z-axis.
z-axis
 1
0
0
 
 
 
i is  0  , j is  1  and k is  0  .
y-axis
0
0
 1
k
j
 
 
 
i
x-axis
For example, the three-dimensional vector
in terms of i, j and k as
–i + 6j –3k
 1 
 6 
  can be written
 3 
 
Vectors written in terms of the unit base vectors i, j and k
are usually said to be written in component form.
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Contents
The magnitude of a vector
Vectors in two and three dimensions
The magnitude of a vector
Multiplying vectors by scalars
Adding and subtracting vectors
Position vectors and coordinate geometry
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Finding the magnitude of a vector
The magnitude (or modulus) of a vector is given by the length
of the line segment representing it.
For example, suppose we have the
vector
A
a
 4
AB = a =  
 2 
B
The magnitude of this vector is written as AB or a .
We can calculate this using Pythagoras’s Theorem.
AB = 42 + 22
= 20
=2 5
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Finding the magnitude of a vector
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Finding the magnitude of a vector
The magnitude of a three-dimensional vector can be found by
applying Pythagoras’s Theorem in three dimensions.
For example, suppose we have the vector
3
AB =  6 
 2 
 
The magnitude of this vector is given by
AB = 32 + 62 + 22
= 9 + 36 + 4
= 49
=7
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Finding the magnitude of a vector
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The distance between two points
If we are given the coordinates of two points and we are asked
to find the distance between them we use Pythagoras’
Theorem in the same way. For example,
Find the distance between the points with coordinates
P(–4, 7, –2) and Q(5, 9, –8).
If d is the distance between the points then, using Pythagoras’
Theorem in three dimensions gives:
d2 = (–4 – 5)2 + (7 – 9)2 + (–2 – –8)2
d2 = 81 + 4 + 36
d2 = 121
 d = 11
In general, if d is the distance between the points (x1, y1, z1)
and (x2, y2, z2) then
d2 = (x1 – x2)2 + (y1 – y2)2 + (z1 – z2)2
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Unit vectors
Remember, if the magnitude of a vector is 1 it is called a unit
vector.
It is possible to find a unit vector parallel to any given vector, a,
by dividing the vector by its magnitude.
The unit vector parallel to the vector a is denoted by a.
So, in general,
a=
a
a
Find a unit vector parallel to b = 4i – j + k
4i  j  k
b=
18
1
=
(4i  j + k )
3 2
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Contents
Multiplying vectors by scalars
Vectors in two and three dimensions
The magnitude of a vector
Multiplying vectors by scalars
Adding and subtracting vectors
Position vectors and coordinate geometry
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Multiplying vectors by scalars
Remember, a scalar quantity can be represented by a
single number.
It has size but not direction.
A vector can be multiplied by a scalar.
For example, suppose the vector a is represented as follows:
a
2a
The vector 2a has the same
direction but is twice as
long.
3
a= 
2
6
2a =  
4
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Multiplying vectors by scalars
 x
 y
In general, if the vector   is multiplied by the scalar k, then:
z
 
 x   kx 
k  y  =  ky 
 z   kz 
   
For example,
 1   2 
2 ×  4  =  8 
 2   4 
   
When a vector is multiplied by a scalar the resulting vector
lies either parallel to the original vector or on the same line.
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Multiplying vectors by scalars
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Contents
Adding and subtracting vectors
Vectors in two and three dimensions
The magnitude of a vector
Multiplying vectors by scalars
Adding and subtracting vectors
Position vectors and coordinate geometry
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Adding vectors
Adding two vectors is equivalent to applying one vector
followed by the other.
5
3
For example, suppose a =   and b =   .
3
 2 
We can represent the addition of these two vectors in the
following diagram:
b
a
8
a +b =  
 1
a+b
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Adding vectors
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Adding vectors
When two or more vectors are added together the result is
called the resultant vector.
 a1 + b1 
 a1 
 b1 


In general, if a =  a2  and b =  b2  , then a + b =  a2 + b2  .
a +b 
a 
b 
 3 3
 3
 3
Given that a = 2i + 6j – k and b = –i + 2j + 7k, find a + b.
a + b = (2 –1)i + (6 + 2)j + (–1 + 7)k
= i + 8j + 6k
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Subtracting vectors
We can think of the subtraction of two vectors, a – b,
as a + (–b).
4
 2 
For example, suppose a =   and b =   .
4
3
a
b
–b
–b
a
a–b
6
a b =  
 1
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Adding and subtracting vectors
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The parallelogram law for adding vectors
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Vector arithmetic
We have seen that vectors can be multiplied by scalars, added
and subtracted. We have also seen that vector addition is
commutative.
We can use this to add and subtract any given multiple of a
vector given in component or column vector form. For example,
Given that a = 2i – 4j + k and b = j + 2k find 3a – 2b.
3a – 2b = 3(2i – 4j + k ) – 2(j + 2k )
= 6i – 12j + 3k – 2j – 4k
= 6i – 14j – k
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Vector arithmetic
6
 4 
Suppose that a =   and b =   .
 1
7
Find vector c such that 2c + a = b.
Start by rearranging the equation to make c the subject.
2c + a = b
2c = b – a
c = 21 (b – a)
6  4 
=


1

7


 5 
 c= 
 4 
1
2
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A grid of congruent parallelograms
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Using vectors to solve problems
We can use vectors to solve many problems involving physical
quantities such as force and velocity.
We can also use vectors to prove geometric results.
For example, suppose we have a triangle ABC as follows:
The line PQ is such that P is the
mid-point of AB and Q is the
mid-point of AC.
B
P
A
Q
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C
Use vectors to show that PQ is
parallel to BC and that the length
of BC is double the length of PQ.
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Using vectors to solve problems
Let’s call AP vector a and AQ vector b.
PQ = a + b
B
=b a
P
BC = 2a + 2b
a
A
= 2b  2a
b
Therefore,
Q
C
= 2(b  a)
BC = 2 PQ
We can conclude from this that PQ is parallel to BC and that
the length of BC is double the length of PQ.
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Collinear points
Three or more points are said to be collinear if they lie on the
same line. For example
Prove that the three points A(–3, 2, 6), B(1, 4, –2)
and C(1, 5, –6) are collinear.
 1  3   4 
AB =  4  2  =  2 
 2  6   8 

  
 1  1   2 
BC =  5  4  =  1 
 6  2   4 

  
 AB = 2BC
Since AB is a scalar multiple of BC the two lines must be
parallel. They also have the point B in common and so the
points A, B and C must be collinear.
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Contents
Position vectors and coordinate geometry
Vectors in two and three dimensions
The magnitude of a vector
Multiplying vectors by scalars
Adding and subtracting vectors
Position vectors and coordinate geometry
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Position vectors
A position vector is a vector that is fixed relative to a fixed
origin O.
For example, suppose the point P has coordinates (4, 6).
P
The position vector of the point P is given
by
4
OP = p =  
p
6
Now, suppose the point Q has
coordinates (3, –2).
O
q
Q
The position vector of the point Q is given
by
3
OQ = q =  
 2 
Write the vector PQ as a column vector.
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Position vectors
To get from the point P to the point Q, we have to go from P to
O…
… and then from O to Q.
P
PQ = q  p
So
 3   4
=  
 2   6 
–p
p
q–p
O
q
Q
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 1 
= 
 8 
We can check this using the vector
diagram.
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Position vectors
In general, if A is the point with coordinates (x1, y1) and B is the
point with coordinate (x2, y2) we can write the position vectors
 x1 
OA = a =  
 y1 
and
 x2 
OB = b =  
 y2 
The vector AB is given by
AB = OB  OA
=b a
 x2  x1 
=

y

y
1
 2
The vector AB can also be written in terms of i and j as
AB = ( x2  x1 )i + ( y2  y1) j
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The mid-point of a line
Returning to our example using the points P(4, 6) and Q(3, –2):
P
Let M be the mid-point of the line PQ.
What is the position vector of the point M?
p
OM = OP + PM
M
= OP + 21 PQ
= p + 21 (q  p)
O
q
Q
= p + 21 q  21 p
= 21 p + 21 q
= 21 (p + q)
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The mid-point of a line
3
4
p =   and q =   so
 2 
6
P
1
2
4+3 
OM =

6
+

2


7
=

4
 
p
1
2
 3 21 
= 
 2 
M
m
O
q
Q
In general, if points A and B have position vectors a and b, then
the position vector of the mid-point of the line AB is given by:
1 (a + b)
2
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