Vectors – Scalar (Components) and Vector Projections                                            

 

 

In a Nut Shell:   The scalar projection of a vector, U,  in the direction of another vector,

V, is just the component of  U along V.   Its symbol is, compV U .

 

                                   

          compv U  =  UV  / |V|  

 

 

 

Note:  The result is a scalar.   UV  / |V|   =  (U) cos θ  (V) / |V|   =  U  cos θ

 

Here   θ   is the angle between the vectors   U  and   V .

 

 

Note:   For two vectors in the   x-y plane the component of   U along V  is   U  cos θ

as shown below.   A similar result holds for vectors in three dimensions, x-y-z.    

 

 

                                  

 

 

 

In a Nut Shell:  The vector projection of a vector, U,  in the direction of another

vector, V,  is just the component of  U  in the direction of  V  times the unit vector

along   V.  The symbol for the vector projection of   U  along  V  is      projV U .

 

 

      projV U  =  [ ( UV ) / |V| ] eV

 

 

Note:  The result is a vector.

 

A unit vector in the direction of  V   is   V / |V|   =  eV

 

So         projV U  =  [ (U  cos θ) ] [ V / |V| ]   =    [ ( UV ) / |V| ] eV

 

 

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