[<< wikibooks] Linear Algebra/Projection
This section is optional; only the last two sections of Chapter Five require this material.
We have described the projection 
  
    
      
        π
      
    
    {\displaystyle \pi }
   from 
  
    
      
        
          
            R
          
          
            3
          
        
      
    
    {\displaystyle \mathbb {R} ^{3}}
   into its

  
    
      
        x
        y
      
    
    {\displaystyle xy}
   plane subspace as a "shadow map".
This shows why, but it also shows that some shadows fall upward.

So perhaps a better description is: the projection of 
  
    
      
        
          
            
              v
              →
            
          
        
      
    
    {\displaystyle {\vec {v}}}
  
is the 
  
    
      
        
          
            
              p
              →
            
          
        
      
    
    {\displaystyle {\vec {p}}}
   in the plane with the property that 
someone standing on 
  
    
      
        
          
            
              p
              →
            
          
        
      
    
    {\displaystyle {\vec {p}}}
   and looking straight up or down sees

  
    
      
        
          
            
              v
              →
            
          
        
      
    
    {\displaystyle {\vec {v}}}
  .
In this section we will generalize this to other projections,
both orthogonal (i.e., "straight up and down") and nonorthogonal.