 Dr Roger Nix (Queen Mary, University of London) - 4.1
CHAPTER 4: VECTORS & COORDIATE SYSTEMS
Vectors
What is a Vector ?
A vector is a quantity with both size and direction. Examples of physical quantities which are vectors
include : force, velocity, acceleration, linear momentum, angular momentum and dipole moment.
By convention, vectors are represented in diagrams by arrows, where the length of the arrow reflects
the size and the direction of the arrow its direction.
a
In text, they are usually written in bold and often underlined : e.g.
or
a
a
The size or magnitude of a vector a is denoted by | a |
Resolving Vectors into Components
In two-dimensions a vector may be resolved into two perpendicular components, conventionally xand y- components parallel to the x- and y- axes respectively.
y
ax = | a | cosθ
a
ay
ay = | a | sinθ
θ
x
ax
Similarly, in three-dimensions a vector may be resolved into three perpendicular components,
conventionally x- , y- and z- components parallel to the x- , y- and z- axes.
z
Note : by convention, θ is
now taken to be the polar
angle between the z-axis and
the vector.
a
az
θ
y
ax
x
ay
It turns out that we are often only interested in the component of the vector quantity parallel to one
particular axis (by convention the z-axis) ; this is again given by a simple trigonometric relationship.
az = | a | cosθ
 Dr Roger Nix (Queen Mary, University of London) - 4.2
Addition of Vectors
If you wish to add together two vectors then this may be done graphically or by separate addition of
the vector components.
Graphical Approach
This is most easily illustrated in two dimensions - consider the vector addition : c = a + b
The diagrams below illustrate two equally-good constructions for carrying out the addition.
y
y
c
c
b
b
a
a
x
x
In the first construction (shown on the left) both vectors are drawn so as to originate from the origin
and a parallelogram is then constructed by taking the two vectors as two adjacent sides of this
parallelogram. The resultant vector c is obtained by linking the origin to the opposite corner of the
parallelogram.
In the second construction (shown on the right) one of the two vectors is translated so that its tail
coincides with the head of the other vector. The resultant vector c is obtained by linking the tail of
the first vector to the head of the second.
Addition of Components
The x- , y- and z- components of both the original vectors first need to be established.
 ax 
 
Suppose : a =  a y 
a 
 z
then if
and
 bx 
 
b =  by  ;
b 
 z
c = a + b
 a x   bx   a x + bx 

    
c = a + b =  a y  +  by  =  a y + by  ,
a  b   a + b 
z 
 z  z  z
i.e.
 c x   a x + bx 

  
c =  c y  =  a y + by 
c   a + b 
z 
 z  z
 Dr Roger Nix (Queen Mary, University of London) - 4.3
Dot Product (Scalar Product) of Two Vectors
For any two vectors, a and b , the dot-product is defined as follows:
a
a • b = | a | | b | cos θ
θ
b
It corresponds to the product of the magnitude of one vector with the component of a second vector
which is parallel to this first vector.
Note that:
•
the dot-product is a scalar quantity (i.e. it only has a magnitude; not a direction)
•
the value of the dot product is zero if the two vectors are perpendicular to one another
(i.e. when θ = 90° , cos θ = 0 )
When do we come across the dot product in chemistry ?
An example, would be in the formula for the energy of interaction between an (electric) dipole and an
electric field. This interaction energy is dependent upon the relative alignment of the dipole with the
field, as expressed in the following dot-product relationship.
ε = –µ•E
 Dr Roger Nix (Queen Mary, University of London) - 4.4
Coordinate Systems
There are two coordinate systems which are commonly used by chemists for describing positions in
three-dimensional space.
Cartesian Coordinates
This is the familiar (x, y, z) set of coordinates. The coordinates correspond to the distances along the
three mutually-perpendicular axes.
z-axis
z = z1
P (x1,y1,z1)
z1
y = y1
O
y-axis
x1
x = x1
ote : by convention the x-, yand z-axes form a "righthanded set" (the positive zaxis corresponds to the
direction that the thumb
points in when the fingers of
the right hand are curved
around from the positive xaxis to the positive y-axis).
y1
x-axis
The coordinates can also be regarded as the x- , y- and z- components of a vector drawn from the
origin to the point of concern.
The distance of a point with coordinates (x, y, z) from the origin is given by the Pythagoras theorem,
d = x2 + y2 + z2
The separation distance of two points (for example, two atoms in a crystal) with coordinates
(x1 , y1 , z1 ) and (x2 , y2 , z2 ) is given by,
d = ( x2 − x1 ) 2 + ( y2 − y1 ) 2 + ( z2 − z1 ) 2
= ∆x 2 + ∆y 2 + ∆z 2
Cartesian coordinates are ideal for the description of structures possessing cubic (or closely related
symmetry), e.g. NaCl, diamond, fcc and bcc metals.
 Dr Roger Nix (Queen Mary, University of London) - 4.5
Spherical Polar Coordinates
This system uses a different set of three coordinates - (r, θ, φ ) . These are defined in the manner
illustrated below.
z-axis
P
r
θ
z
O
y-axis
x
φ
B
y
x-axis
The permitted ranges of the three coordinates (so as to be able to uniquely define each point in space
with a single set of coordinates) are:
r
:
0 → ∞
(i.e. positive values only)
θ
:
0 → 180°
(i.e. 0 → π radians)
φ
:
0 → 360°
(i.e. 0 → 2π radians)
The spherical polar coordinates are easily derived from the Cartesian coordinates using basic
trigometry:
r=
x2 + y 2 + z 2
tan θ =
x2 + y2
z
and
tan φ =
y
x
The spherical polar coordinate system is a much more convenient coordinate system for describing
functions (or objects) with spherical aspects of symmetry - the main chemical example being the
atomic orbitals (wavefunctions).