Resolution of Vectors
Splitting up of a vector into two or more than two parts such that sum of these parts give the original vector, is known as resolution of a vector.
The parts of a vector obtained from resolution are known as components of the vector.
It is a process just opposite of the addition of two or more vectors.
So, if 
, then 
is the resultant.
Conversely, 
i.e. the vector can be split up so that the vector sum of the split parts equals the original vector . These split parts are known as components of and this process is known as resolution.
Rectangular components of a Vector:
When a vector is resolved in two mutually perpendicular direction, the components thus obtained are known as rectangular components of the vector.
In physics, resolution gives unique and mutually independent components only if the resolved components are mutually perpendicular to each other. Such a resolution is known as rectangular or orthogonal resolution and the components are called rectangular or orthogonal components.
These are also called projections of the given vector on the axes.
To get rectangular components of a given vector (say 
), follow these steps
Step 1: Draw the rectangular axes (X- axis and Y- axis) which are intersecting at the tail of the given vector.
Step 2: Drop perpendiculars on these axes from the head of the vector.
Step 3: Join the foot of these perpendiculars to the tail of the given vectors. These lines thus formed along the axes represent the components of the given vector. Be sure to place arrowheads on these components at the foot of the perpendiculars on the respective axes, to indicate their direction.
Consider a vector that lies in xy plane as shown in figure,
The quantities Ax and Ay are called x– component and y– component of the vector respectively. Ax is itself not a vector but 
is a vector and so is 
Ax = A cos θ and Ay = A sin θ
Its clear from above equation that a component of a vector can be positive, negative or zero depending on the value of θ. A vector can be specified in a plane by two ways:
- as its magnitude and the direction q it makes with the x – axis; or
- as its components Ax and Ay.
Note : If θ = 0o , Ax = A and Ay = 0 i.e.
Components of a vector perpendicular to itself is always zero.
Resolution of Vectors (Video)
(In Hindi + English Mix)
Resolution of Vectors in Three Dimensions
In three dimension, a vector can have three components along along x-,y- and z-axis. These are 
, 
and 
respectively. So, the vector can be written as :
The magnitude of the vector is given by 
If α , β and γ are the angles subtended by the vector with x-, y- and z- axis respectively, then
where cos α, cos β and cos γ are termed as Direction Cosines of a given vector .
Also, for any vector in three dimension
cos2 α + cos2 β + cos2 γ = 1
Note : Rectangular Component of any vector along another direction, which is at an angular departing θ, is the product of the magnitude of the vector and cosine of the angle between them (θ). Therefore the component of is A cos θ.
Example: A force of 30 N is acting at an angle of 60º with the y axis. Determine the components of the force.
Solution :
Fx = F sin 60º
Example: A mass of 2 kg lies on an inclined plane making 30̊ with horizontal. Resolve its weight along and perpendicular to the plane. (Assume g = 10 m/s2)
Solution. Component along the plane = mg sin 30°
=20 sin 30° = 10 N
Component perpendicular to the plane = mg cos30°
= 20 cos 30°
Example: 
. When a vector 
is added to , we get a unit vector along x-axis. Find the value of . Also find its magnitude.
Solution:
Example: Vector , and 
have magnitude 5, 5√2 and 5 respectively. Directions of , and are towards east, North-East and North respectively. Find magnitude and direction of the resultant of these three vectors.
Solution: Consider 
and 
as unit vectors along East and North respectively.
