Three Dimensional Force Systems and Dot Product
There are three different ways to define a vector in space. I refer to these as the three
angle method, two point method and the two angle method. Each is discussed briefly
below.
Three Angle Method
In this case the force is defined by the three angles x , y and z which its direction
makes with the three positive coordinate axis. These are shown in the picture below.
z
F
y
θy
θz
θx
x
The direction cosines l, m and n are the cosines of these angles and given by the formulas
l = cos x , m = cos y , n = cos z
l 2 + m 2 + n 2 =1
The scalar components of F are F x , F y and F z and are given as
F x =F l , F y = F m ,
F z =F n
The vector F has the form
F = F x i + F y j + F z k = F ( li + mj + nk )
A unit vector n in the direction of F is found to be
n = (li + mj + nk)
Two Point Method
In this case the force is defined by two points A and B along its line of action. This is
shown in the picture below.
z
F
B ( x 2, y 2, z 2 )
A ( x 1 , y1 , z 1 )
y
x
The force is given as F = Fn where the unit vector n in the direction of the force can be
found from the coordinates of the two points. The vector r from A to B is
r = (x 2 – x 1) i + (y 2 – y 1) j + (z 2 – z 1) k
| r | = √ (x 2 – x 1)2 + (y 2 – y 1)2 + (z 2 – z 1)2
n=r/|r|
Two Angle Method
In this case the force is defined by two angles and . These are shown in the picture
below. The force F xy is the component of F in the x – y plane. The angle is measured
from the positive x axis to the F xy direction (0 <  < 360). The angle  is measured from
the x – y plane to the F direction (-90 <  < 90).
z
F
Fz

θ
F xy
y
x
By making right triangles the magnitudes are related by
F xy = F cos 
 F x
,
F z = F sin 

= F xy cos  F cos  cos  F y = F xy sin  F cos  sin 
Dot Product
The dot product is a vector operation defined between two vectors P and Q. These
two vectors are shown where the angle between their positive directions is 
(0 <  < 180).
Q
θ
P
Whereas the cross product of two vectors produced a new vector, the dot product
produces only a scalar. It is define as
P . Q = PQ cos
If two vectors are perpendicular their dot product is zero. The dot product of unit vectors
i, j and k are easily found as
i .i =1 , j .j =1 , k.k=1 , i .j =0 , i .k=0 , j .k=0 ,
etc.
In most cases the angle is not known but the components of the forces are. If the
rectangular components of P and Q are known then the dot product can be alternatively
found as
P . Q = ( P x i + P y j + P z k) . (Q x i + Q y j + Q z k) = P x Q x + P y Q y + P z Q z
The angle between two vectors can then be found as
cos-1 [(P . Q )/(PQ)] = cos-1 [(P x Q x + P y Q y + P z Q z)/(PQ)]
The dot product can also be used to find the projection of a vector in a specified
direction. In the figure below n is a unit vector in a specified direction and P is a vector.
P
n
α
Pn
The projection Pn of P in the direction of the dashed line is given as
Projection of P = Pn = P cos  = P . n
If you want the projection of a force in a certain direction you must dot the force into a
unit vector in that direction.