VECTORS
Vectors are quantities that are completely described by a
scalar magnitude and a direction. Displacement, velocity,
acceleration and force can all be vectors – they can be
described fully by their magnitude and direction.
Vectors are often represented by a directed line segment.
B
This vector could be represented
as: a, a or AB.
A
A bold letter is used to represent vectors in text books and on
exam papers. When writing by hand, the convention is to
underline the letter.
1 of 27
© Boardworks Ltd 2005
Column and component form
A vector can be represented in component form as ai + bj
in two dimensions, or ai + bj + ck in three dimensions.
The vectors i, j and k are unit vectors in the positive
directions of the x, y and z axes respectively.
A vector can also be represented in column form
a

 a
as  or  b .
b

c
 
2 of 27
© Boardworks Ltd 2005
Position vectors
The position vector of a point A is the vector OA where
O is the origin.
r is often used to denote a position vector.
For example, if OA = 3i + 2j – k, we might say
r = 3i + 2j – k.
3 of 27
© Boardworks Ltd 2005
Magnitude of a vector
The magnitude of a vector is the size of the vector. When
representing a vector by a directed line segment, the
magnitude of the vector is represented by its length.
The magnitude of a vector is equal to a .
The magnitude of a vector is found using Pythagoras’
Theorem.
If x = ai + bj then x = a 2  b 2
x
b
Similarly, in three dimensions,
if x = ai + bj + ck
a
then x = a 2  b 2  c 2
4 of 27
© Boardworks Ltd 2005
Magnitude questions
Find the magnitude of the following vectors:
a) 3i + 2j
b) 4i – 6j + k
c) -i + 3j – 5k
5 of 27
d)
 - 3

÷

÷

÷
 5
e)
 2

÷

÷
 -1÷

÷
 3÷


© Boardworks Ltd 2005
Magnitude solutions
a) Magnitude = 32  22  9  4  13
b) Magnitude = 42  ( -6)2  12  16  36  1  53
2
2
2
(
1)

3

(
5)
 1  9  25  35
c) Magnitude =
d) Magnitude = ( -3)2  52  9  25  34
2
2
2
e) Magnitude = 2  ( -1)  3  4  1  9  14
6 of 27
© Boardworks Ltd 2005
Unit vectors
A unit vector is a vector of magnitude 1.
If v is a vector then the corresponding unit vector is
represented by v̂ .
A unit vector in the direction of a given vector is found by
v
dividing the vector by the magnitude: vˆ =
v
7 of 27
© Boardworks Ltd 2005
Unit vector questions
Find a unit vector in the direction of the following vectors:
a) 4i – 2j
b) 3i – j + 4k
c) –3i + 5j – 2k
d)
e)
8 of 27
 3
 ÷
 ÷
 ÷
6
 - 4

÷

÷
 0÷

÷
 3÷


© Boardworks Ltd 2005
Unit vector solutions
a) Magnitude  20  2 5 unit vector 
1
 4i - 2 j
2 5
1
b) Magnitude  26 unit vector 
3i - j  4k 
26
1
c) Magnitude  38 unit vector 
 -3i  5 j - 2k 
38
d) Magnitude  45  3 5  unit vector 
3

 1 
5
 3 5

6
  2





5

 3 5
 -4 
 5
e) Magnitude  25  5 unit vector   0 


3 
 5 
9 of 27
© Boardworks Ltd 2005
Multiplication of a vector by a scalar
To multiply a vector by a scalar, multiply each component of
the vector by the scalar.
3(2i – 5j + k) = 6i – 15j + 3k
 0  0 
-5  -6    30 
 2   -10 
  

If the scalar is greater than 1, the resulting vector is larger
than the original vector and parallel to it.
If the scalar is less than 1, the resulting vector is smaller
than the original vector and parallel to it.
10 of 27
© Boardworks Ltd 2005
Parallel vectors
For vectors to be equal, the i, j and k components must be
equal.
One vector is the negative of another if each of their
corresponding components have opposite signs.
Two vectors are parallel if one is a scalar multiple of the other:
3i – 3j + 4k and –9i + 9j – 12k are parallel vectors since
–9i + 9j – 12k = –3(3i – 3j+ 4k)
11 of 27
© Boardworks Ltd 2005
Addition and subtraction
Adding and subtracting vectors in component form is
simply a case of adding and subtracting the i, j and k
components separately.
Add the following vectors: 3i + 5j – k, 2i + 3k and –4i + 3j + k.
(3i + 5j – 2k) + (2i + 3k) + (–4i + 3j + k) = i + 8j + 2k
 3  5
 -2 
Add the following vectors :  -2  ,  -3  and  3 
 1  0 
 4
   
 
 3   5   -2 
 -2    -3    3  
     
 1  0   4 
     
12 of 27
 35-2 
 -2 - 3  3  


 1 4 


 6
 -2 
 
 5
 
© Boardworks Ltd 2005
A unit vector is a vector of length 1. Unit vectors along Cartesian (x, y) axes are usually
denoted by i and j respectively.
(0,1)
j
O
i
You can write any two-dimensional vector in the form ai + bj
(1,0)
C
5i + 2j
2j
A
Example
Draw a diagram to represent the vector -3i + j
B
5i
𝐴𝐶 = 𝐴𝐵 + 𝐵𝐶
𝐴𝐶 = 5𝒊 + 2𝒋
-3i + j
j
-3i
13 of 27
© Boardworks Ltd 2005
Examples
(a)
Find 2a – b in terms of i and j
2𝒂 + 𝒃 = 2(5𝒊 + 2𝒋) − (3𝒊 − 4𝒋)
2𝒂 + 𝒃 = 10𝒊 + 4𝒋 − (3𝒊 − 4𝒋)
2𝒂 + 𝒃 = 10𝒊 + 4𝒋 − 3𝒊 + 4𝒋
2𝒂 + 𝒃 = 7𝒊 + 8𝒋
b) Find the magnitude of the vector 3i – 7j
𝒗 =
3i
32 + (−7)2
Put in the values from
the vectors and
𝒗 = 7.62 (3sf)
calculate
𝒗 = 58
14 of 27
-7j
3i - 7j
© Boardworks Ltd 2005
Example
Find the angle between the vector -4i + 5j and the positive x-axis.
 Draw a diagram
y
Opp
𝑂𝑝𝑝
𝑇𝑎𝑛𝜃 =
𝐴𝑑𝑗
5j
51.3°
θ
-4i
Adj
x
5
𝑇𝑎𝑛𝜃 =
4
𝜃 = 51.3°
𝐴𝑐𝑡𝑢𝑎𝑙 𝑎𝑛𝑔𝑙𝑒 = 128.7°
15 of 27
The angle we want is
between the vector and
the positive x-axis
 Subtract θ from 180°
6D
© Boardworks Ltd 2005
The velocity of a particle is a vector in the
direction of motion. The magnitude of the
vector is its speed. Velocity is usually
represented by v.
Example
A particle is moving with constant velocity
given by:
v = (3i + j) ms-1
Find:
a)The speed of the particle
b)The distance moved every 4 seconds
16 of 27
3i + j
j
(a) Finding the speed
3i
 The speed of the particle is the magnitude of the
vector  Use Pythagoras’ Theorem
𝑣 = 32 + 12
𝑣 = 3.16𝑚𝑠 −1
(b) Finding the distance travelled every 4 seconds
 Distance = Speed x Time
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 𝑠𝑝𝑒𝑒𝑑 × 𝑡𝑖𝑚𝑒
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 3.16 × 4
Sub in values (use the
exact speed!)
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 12.6𝑚
© Boardworks Ltd 2005