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Distance from a point to a plane

<p>In <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>, the distance from a point to a plane is the distance between a given point and its <a href="/facts/Orthogonal_projection/sElXGkxD">orthogonal projection</a> on the plane, the <a href="/facts/Perpendicular_distance/P0aEswE4">perpendicular distance</a> to the nearest point on the plane.
</p><p>It can be found starting with a <a href="/facts/Change_of_variables/vjWdXCDF">change of variables</a> that moves the origin to coincide with the given point then finding the point on the shifted <a href="/facts/Plane_(mathematics)/tAUtRFcn">plane</a> 
  
    
      
        a
        x
        +
        b
        y
        +
        c
        z
        =
        d
      
    
    {\displaystyle ax+by+cz=d}
  
 that is closest to the <a href="/facts/Origin_(mathematics)/Lm26tRCB">origin</a>. The resulting point has <a href="/facts/Cartesian_coordinate/lkTqvMmB">Cartesian coordinates</a> 
  
    
      
        (
        x
        ,
        y
        ,
        z
        )
      
    
    {\displaystyle (x,y,z)}
  
:
</p>

x
          =
          
            
              
                a
                d
              
              
                
                  a
                  
                    2
                  
                
                +
                
                  b
                  
                    2
                  
                
                +
                
                  c
                  
                    2
                  
                
              
            
          
          ,
          
          
          
            y
            =
            
              
                
                  b
                  d
                
                
                  
                    a
                    
                      2
                    
                  
                  +
                  
                    b
                    
                      2
                    
                  
                  +
                  
                    c
                    
                      2
                    
                  
                
              
            
            ,
            
            
            
              z
              =
              
                
                  
                    c
                    d
                  
                  
                    
                      a
                      
                        2
                      
                    
                    +
                    
                      b
                      
                        2
                      
                    
                    +
                    
                      c
                      
                        2
                      
                    
                  
                
              
            
          
        
      
    
    {\displaystyle \displaystyle x={\frac {ad}{a^{2}+b^{2}+c^{2}}},\quad \quad \displaystyle y={\frac {bd}{a^{2}+b^{2}+c^{2}}},\quad \quad \displaystyle z={\frac {cd}{a^{2}+b^{2}+c^{2}}}}
  
.
<p>The distance between the origin and the point 
  
    
      
        (
        x
        ,
        y
        ,
        z
        )
      
    
    {\displaystyle (x,y,z)}
  
 is 
  
    
      
        
          
            
              x
              
                2
              
            
            +
            
              y
              
                2
              
            
            +
            
              z
              
                2
              
            
          
        
      
    
    {\displaystyle {\sqrt {x^{2}+y^{2}+z^{2}}}}
  
.
</p>