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Resolving a Vector into Two Perpendicular Components
A vector quantity (eg force, acceleration etc) has its full effect in a particular direction but it also has reduced effects in other directions.  
   
The effect of a vector in a direction not along its own line of action is called a component of the vector.  
   
The process of finding the magnitudes of the components of a vector is called resolving the vector into its components.  
   
  Consider a vector v acting in some arbitrarily chosen direction, as shown here.    
We know (see adding vectors) that this vector could be made by adding together 2 other vectors.   
These "two other vectors" could be any pair out of many possible pairs but for both practical and theoretical reasons it is often useful to choose one vertical and one horizontal vector.   
Theoretical reason: two perpendicular vectors can be considered to be independent of each other.     
A vertical vector has no horizontal effect and vice versa.    
(Throw a ball vertically upwards... if there's no wind, it will come back down on your head, no matter how fast you throw it !)   
Practical reason: we are often dealing with situations in which gravity has an effect... gravity acts vertically.   
     
The diagram here represents a sort of "vector addition in reverse".    
By completing a rectangle around the original vector we can find the magnitudes of a pair of vertical and horizontal vectors which would have the same overall effect as v.   
   
These are called the vertical and horizontal components of v.   
   
These are called the vertical and horizontal components of v.   
   
   
These statements show mathematically what was stated above, namely that a vector has zero effect in a direction at 90 to its own line of action (cos90 = 0)   
   
The motion of projectiles in a gravitational field is a situation in which the process of resolving a velocity into its vertical and horizontal components is very useful.   
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