SCALARS AND VECTORS
Definition: A scalar is a number or a quantity that has only a magnitude and no
direction. Examples include temperature, speed, mass etc.
The primary school algebra can be called scalar algebra as it dealt with scalars only.
Definition: A vector is a list of numbers. It can be taken as a point in space in which
case the list of numbers is a way of identifying this point. When a vector is considered
as a magnitude and direction, one can think of directed arrow from the origin of a
cartesian plane to an end point being given by the list of numbers.
The numbers are called components of the vector.

In this case, the vector will be represented by either of the symbols: a, a or 𝑎̃.
y
a
12
2
5

x
An example of a vector is a  (5,12) which is represented in the above diagram
graphically. Think of it as an arrow in the x  y plane pointing from the origin to the
point (5,12). This having a list of only two numbers is a 2  dimensional vector and is
said to be on a plane or in ℝ2 where ℝ is the set of real numbers.
A 3-dimensional vector would be a list of three numbers with the space called ℝ3 and
an n-dimensional vector would be a list of n numbers and the space is ℝ𝑛 .
Magnitude and Direction
The magnitude of a vector is the distance from its ‘head’ to its tail or the origin or in
other words its length. Using the Pythagorean theorem, the length of the above vector


a  (5,12) denoted by | a | is given by

| a | 5 2  122  169  13 .
The magnitude is a scalar.
Vector scalar multiplication
when we multiply a vector by a scalar, each of its components is multiplied by the


scalar. For example if k is a scalar and a  ( x, y) is a vector, then k a  k ( x, y)  (kx, ky).
Definition: A unit vector is a vector of magnitude 1. It gives the general direction of
any given vector.


The unit vector of any vector a is given by
a

. The unit vector of the vector above is
|a|
therefore equal to
(5,12)  5 12 
 , .
13
 13 13 
 5 12 
, .
 13 13 
Vector (5,12) is therefore said to be in the direction of 
In ℝ2 , the unit vector in the direction if the x  axis is denoted by 𝒊 = (1,0) and that
along the y  axis by 𝒋 = (0,1).
In ℝ3 we have the x, y and z axes with the corresponding unit vectors, 𝒊 = (1,0,0),
𝒋 = (0,1,0) and 𝒌 = (0,0,1)
Vector Addition and Subtraction
A vector can be added to another or subtracted from another if and only if the two are
of the equal dimensions. In carrying out these operations, corresponding vectors are
considered. The sum in each case is another vector.


For example if a  (2,5) and b  (7,11) , then


b  a  (7, 11)  (2, 5)
 (7  2, 5  11)
 (9, 16)
and


b  a  (7,11)  ( 2,5)
 (7  2, 11  5)
 (5,6)
This concept can be used to describe displacement.
Definition: The zero vector is the vector (0,0) ∈ ℝ2 , (0,0,0) ∈ ℝ3 and (0,0, ⋯ ,0) ∈ ℝ𝑛
whose magnitude is zero and has no direction.
Definition: Two vectors are said to be equal if they have the same direction and
magnitude. Graphically, the two must be parallel.
In terms of components, two vectors are equal if the corresponding components are


equal. For example vectors u  (a, b) and v  (c, d ) are equal if a  c and b  d .
Exercise: Find the values of the unknown in:
1.  (1,2)  (3,6)
2. (1,2)   (3,4)   (1,1)
3.  (1,2)  (3,4)
4.  (5i  j )   ( j  k )  k  5i  3 j  k
Linear Combinations
Definition: A vector 𝒖 is said to be a linear combination of vectors a and b if it can be
expressed in terms of a and b using some scalars 𝜆 and 𝜇 i.e 𝒖 = 𝜆𝒂 + 𝜇𝒃.
Any vector in 𝒖 = (𝑎, 𝑏) ∈ ℝ2 can be written as a linear combination of the unit vectors 𝒊
and 𝒋. Thus
𝒖 = (𝑎, 𝑏) = (𝑎, 0) + (0, 𝑏)
= 𝑎(1,0) + 𝑏(0,1)
= 𝑎𝒊 + 𝑏𝒋
In ℝ3 we have 𝒖 = (𝑎, 𝑏, 𝑐) = (𝑎, 0,0) + (0, 𝑏, 0) + (0,0, 𝑐)
= 𝑎(1,0,0) + 𝑏(0,1,0) + 𝑐(0,0,1)
= 𝑎𝒊 + 𝑏𝒋 + 𝑐𝒌
Examples:
1. Is (2,1) a linear combination of (1,0) and (−1, −1)?
2. Is (0,0,0) a linear combination of (2,1,1), (1,0,2) and (−1, −1,1)?
3. Check whether or not (0,0,1) is a linear combination of (2,1,1), (1,0,2) and
(−1, −1,1).
Properties of vectors
Let a, b and c be vectors and 𝜆 and 𝜇 be scalars then;
1. 𝒂 + 𝒃 = 𝒃 + 𝒂 (Commutativity law)
2. 𝒂 + (𝒃 + 𝒄) = (𝒂 + 𝒃) + 𝒄 (Associativity law)
3. 𝒂 + 𝟎 = 𝟎 + 𝒂 = 𝒂 (0 is an additive identity)
4. 𝒂 + (−𝒂) = −𝒂 + 𝒂 = 𝟎 (-a is an additive inverse)
5.
 (a  b)  a  b 
 Distributive laws
a(   )  a  a 
6. (𝜆𝜇)𝒂 = 𝜆(𝜇𝒂)
7. 1(𝒂) = (𝒂)1 = 𝒂. 1 is a multiplicative identity.
8. |𝜆𝒂| = |𝜆||𝒂|