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Resolution of Vectors

The process of splitting a vector is called resolution of a vector. The parts obtained after resolution are known as components of the given vector.

Consider two non-zero vectors and in a plane. Let be any other vector in this plane. Through the tail (P) of , draw a straight line parallel to .

Similarly, draw a straight line, parallel to , through the terminal point (Q) of . Let both the lines intersect at C.
Applying triangle law of vectors: = +
As per the geometrical construction, = λ
where λ is a real number. In the given case, λ is positive which indicates that is in the direction of . If λ were negative, then would have been opposite to .
where μ is another real number

So, has been resolved along and.
It may be noted that determine μ and λ unambiguously. The converse is also true, i.e., each vector in a plane is completely described or determined by a pair of real numbers λ and μ . The uniqueness of the resolution procedure is proved below:
Let us assume that there are two ways of resolving along and such that

(λ - λ') = (μ' - μ )
But and are different vectors.
So, the above equation is satisfied only if = = .
Thus there is one and only one way in which a vector can be resolved along and . However, it may be pointed out here that a vector may be resolved into an infinite of components. The reverse process, i.e. the sum of the components will of course yield only the given vector.

In the figure, the resolution of a position vector has been shown.
Applying parallelogram law of vectors, we can prove that λand λare actually the components of .
A1 = Axi, A2 = Ayj
where Ax and Ay are real numbers.
Thus, A = Axi+Ayj.

The quantities Ax and Ay are called x, y components of the vector A. Note that Ax is itself not a vector, but Axi is a vector, and so is Ayj. Using simple trigonometry, we can express Ax and Ay in terms of the magnitude of a and the angle θ it makes with the x-axis:

Ax = A cos θ
Ay = A sin θ
As is clear from the above equation, a component of a vector can be positive, negative or zero depending on the value of θ.

Now, we have two ways to specify a vector A in a plane. It can be specified by:
  1. Its magnitude A and the direction θ it makes with the x-axis; or
  2. its components Ax and Ay
If A and θ are given, Ax and Ay can be obtained using the above equation. A and θ can be obtained as follows:
Ax2+Ay2 = A2cos + A2sin = A2.

So far we have considered a vector lying in an x-y plane. The same procedure can be used to resolve a general vector A into three components
along x-, y-, and z-axes in three dimensions. If α, β, and are the angles between A and x-, y- and z-axes respectively.
Ax = A cosα,
Ay = Acosβ,
Az = A cosγ

In general, we have
A = Axi+Ayj+Azk

The magnitude of vector A is

A position vector ‘r; can be expressed as
r = xi+yj+zk

where x, y, and z are the components of ‘r’ along x, y, z-axes respectively.

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