Vector Components

Components of a Vector: 
From the figure given below we can write 
                                          
It means B,C  and D vectors are the components of vector A 

                                     . 

Note that we can also write A =X +Y. So vectors X and Y are also components of vector A. It implies that we can draw any number of set of components in any desired direction.

For resolving a vector A along two directions making angles α and with it as shown in figure given below, we use the following:-
 
                                     

                               Aα = (A sinβ)/(sin(α + β))

Perpendicular Components

Representation of any vector lying in the x – y plane, as shown in figure given below, as the sum of two vectors, one parallel to the x-axis and the other parallel to y-axis, is extremely useful in physical analysis because both have mutually independent effects. These two vectors are labeled A and A. These are called the perpendicular or rectangular component of vector A and are expressed as: 
                             

If magnitude and direction of vector A are known then A_x = A cosθ and A_y = A sinθ. Hence we can write 

                              A_x² + A_y² = A₂ and tanθ = A_y/A_x

Now, it should be clear that if we have to add 30 vectors, we will resolve each of these 30 vectors in rectangular components in any x and y direction. Then simply add all the components in the x direction and all the components in the y direction. Adding these two resultant perpendicular components will give us the final resultant.

Illustration:

A vector quantity of magnitude L acts on a point A along the direction making an angle of 45⁰ with the vertical, as shown in the figure given below. Find the component of this vector in the vertical direction?
                                        

Solution:

The component of the vector in the vertical direction will be

                                       L*(cos ∏/4) = L/√2.

Learning components of a vector is very important from IIT JEE, AIEEE and other engineering exams perspective. Most of the Physics is based on usage of vectors and components of vectors. Vectors and their components are also useful in trigonometry and mathematics as a whole. This is a very useful concept and should be learned thoroughly at the beginner’s level so that it paves a way for great understanding of statistics, physics and mathematics at higher levels.

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