Resolution of a vector into its components is the process to determining a set of vectors, whose resultant is the given vector. Each vector in that set is called a component of the given vector.
Prove: Let us consider 
(= 
) is resolute along the line OA and OB creates angles 
and 
with OC respectively. From point C, CD and CE are drawn parallel to OB and OA respectively. OD and OE represent 
and 
respectively.
From 
ODC, 
COD = , OCD = , then ODC = 1800– ( + ).
From trigonometry, we can write 
= 
= 
Or, = = 
Or, 
= 
= 
So, P = 
and Q = 
.
Example: A bob of mass 2kg is suspended freely from the celling by a string. Now a horizontal force F is applied on the bob and the string creates angle 
with vertical. What is the value of tension acts on the string and F if the bob is at rest?
The angle between tension T and F is 
, the angle between F and weight of the bob (Mg = 20N) is 
and the angle between Mg and T is 
. Using Lami’s theorem we get, 
= 
= 
.
Now =
Or, 
= 
F = 20tan = 
.
Again = or, T = = 
.
Resolution of a vector into two rectangular components:
Let us consider OX and OY are two mutually perpendicular axes, where O is the origin. OB represents creates angle θ with +ve x axis. Let BA and BC are the perpendiculars on OX and OY axis respectively. and are the components of and represented by OA and OC respectively. I.e. = + .
The magnitudes of and are represented as P = R
[
= ]
and Q = R
[
= ]
Therefore, R = 
and tan
= 
.
Vector addition by resolution: Let us consider XX/ and YY/ are two perpendicular axes, where O is the origin. 
, 
and 
create angles 
, 
and 
respectively with +ve x axis.
The components of , and along X axis are respectively Px = Pcos, Qx = Qcos(
– ) = – Qcos and Rx = Rsin(
– ) = – Rcos.
The components of , and along Y axis are respectively Py = Psin, Qy = Qsin( – ) = Qsin and Ry = Rcos( – ) = – Rsin.
Let the resultant along X axis is Fx = Px + Qx + Rx (let Fx is along + ve x axis) and along Y axis is Fy = Py + Qy + Ry (let Fy is along + ve Y axis).
So, the resultant of Fx and Fy is F (= 
). If 
creates angle θ with + ve x axis then tan
= 
.
