Lesson Notes 13-1
Vector Basics Part I
If you travel 4 kilometers north and 3 kilometers east, how far have you traveled?
A vector is a quantity that has size (magnitude) and direction. Examples of vectors are
displacement and velocity.
A scalar is a quantity that has size but no direction. Examples of scalars are distance and
speed.
Vectors are represented using directed line segments where the length of the line
represents the size of the vector quantity and the direction on the line shows the direction
of the vector. Consider the points A(2, 3) and B(5, 7) on the Cartesian plane:
This vector can be represented in a variety of ways:
1. Column vector form –
2. Unit vector form –
There are three unit vectors:
1. In the direction of the x-axis is i.
2. In the direction of the y-axis is j.
3. In the direction of the z-axis is k.
Example 1: Write the following in unit vector form: a =  2 
 
−3
Example 2: Write -2i + 6k in column vector form.
The magnitude of vector AB is the length of the vector and is denoted by AB.
Magnitude is found by using Pythagoras’ Theorem.
 a
If AB =   = ai + bj then AB =
 b
If
 a
 
AB =
AB =  b  = ai + bj+ ck then
 
 c
Example 3: Find the magnitude of these vectors
 3
 
a. −2
 2
 
 4
 
b.  9 
a2 + b2 .
a 2 + b 2 + c2 .
Two vectors are equal if they have the same direction and the same magnitude. If two
vectors have the same magnitude but different directions they are called negative
vectors. Two vectors are parallel if one is a scalar multiple of the other. So, AB and
RS are parallel if AB = kRS where k is a scalar quantity. This can also be written as a =
kb.
Example 4:
Example 5: For what values of t and s are these two vectors parallel?
m = 3i + tj – 6k and n = 9i – 12j + sk
Position vectors are vectors giving the position of a point, relative to a fixed origin, O.
The point P with coordinates (x, y) has position vector
 x
OP =   = xi + yj
 y
To find the resultant vector AB between two points A and B we can subtract the
position vector of A from the position vector of B.
Example 6: Points A and B have coordinates (-3, 2, 7) and (2, 7, 0). Find the vector AB.
Example 7: Given that
 2
 0 
  and


XY =  1 
XZ =  −10
 


 −3
 −1 