Geometric interpretation of Factor Analysis parameters for 3 respondents to question "a". The "answer" is represented by the unit vector 
  
    
      
        
          
            z
          
          
            a
          
        
      
    
    {\displaystyle \mathbf {z} _{a}}
  
, which is projected onto a plane defined by two orthonormal vectors 
  
    
      
        
          
            F
          
          
            1
          
        
      
    
    {\displaystyle \mathbf {F} _{1}}
  
 and 
  
    
      
        
          
            F
          
          
            2
          
        
      
    
    {\displaystyle \mathbf {F} _{2}}
  
. The projection vector is 
  
    
      
        
          
            
              
                
                  z
                
                ^
              
            
          
          
            a
          
        
      
    
    {\displaystyle {\hat {\mathbf {z} }}_{a}}
  
 and the error 
  
    
      
        
          
            ε
          
          
            a
          
        
      
    
    {\displaystyle {\boldsymbol {\varepsilon }}_{a}}
  
 is perpendicular to the plane, so that 
  
    
      
        
          
            z
          
          
            a
          
        
        =
        
          
            
              
                
                  z
                
                ^
              
            
          
          
            a
          
        
        +
        
          
            ε
          
          
            a
          
        
      
    
    {\displaystyle \mathbf {z} _{a}={\hat {\mathbf {z} }}_{a}+{\boldsymbol {\varepsilon }}_{a}}
  
. The projection vector 
  
    
      
        
          
            
              
                
                  z
                
                ^
              
            
          
          
            a
          
        
      
    
    {\displaystyle {\hat {\mathbf {z} }}_{a}}
  
 may be represented in terms of the factor vectors as 
  
    
      
        
          
            
              
                
                  z
                
                ^
              
            
          
          
            a
          
        
        =
        
          ℓ
          
            a
            1
          
        
        
          
            F
          
          
            1
          
        
        +
        
          ℓ
          
            a
            2
          
        
        
          
            F
          
          
            2
          
        
      
    
    {\displaystyle {\hat {\mathbf {z} }}_{a}=\ell _{a1}\mathbf {F} _{1}+\ell _{a2}\mathbf {F} _{2}}
  
. The square of the length of the projection vector is the communality: 
  
    
      
        
          |
        
        
          
            
              
                
                  z
                
                ^
              
            
          
          
            a
          
        
        
          
            |
          
          
            2
          
        
        =
        
          h
          
            a
          
          
            2
          
        
      
    
    {\displaystyle |{\hat {\mathbf {z} }}_{a}|^{2}=h_{a}^{2}}
  
. If another data vector 
  
    
      
        
          
            z
          
          
            b
          
        
      
    
    {\displaystyle \mathbf {z} _{b}}
  
 were plotted, the cosine of the angle between 
  
    
      
        
          
            z
          
          
            a
          
        
      
    
    {\displaystyle \mathbf {z} _{a}}
  
 and 
  
    
      
        
          
            z
          
          
            b
          
        
      
    
    {\displaystyle \mathbf {z} _{b}}
  
 would be 
  
    
      
        
          r
          
            a
            b
          
        
      
    
    {\displaystyle r_{ab}}
  
 : the (a,b) entry in the correlation matrix. (Adapted from Harman Fig. 4.3)[9]