Tangent Planes and Normal Lines
If    a b we can write this equation as  a b    
or the level surface  a   b   where  a  b   a b  
Similarly we could do the same thing with an equation like
        and rewrite it as          
where  a   b         
Note that  a   b is normal to the level surface  a  b  
at the point a   b. Therefore  is a vector normal to the surface
   a b where  a  b   a b  
Def: The tangent plane to a surface at a point is a plane through this
point which is perpendicular to the normal line at this point.
Example: Consider the surface      at the point a  b
      this is a 2D vector. It is not normal to the
surface at a  b At this point,   
However it is normal to the level curve      at the point
a b Similarly, at the point a  b    and  is normal to
the curve      at the point a b
4j
y
2
2 x
4i
To get a vector normal to the surface     
we need to consider  a   b       and get 
which is      and at the point a  b this vector
is a   b
(2,0,4)
z
4i-z
x
Example: Find an equation of the tangent plane and find parametric
equations for the normal line to the surface     at the point a  b
 a   b     
         and  a  b       
       and  is normal to the plane 
the equation of the plane has the form
               the equation is
4       
The parametric equations for the normal line can be derived from





which means the parametric equations are



               
               
Example: Find the equation of the tangent plane to the surface
  È   at the point a  b
È      
 a   b  È    
 





È  
È  
 a b 


    


Use the vector a   b
The tangent plane is of the form       
 9+16  25=0  the plane is       
Or we can do the following:
     
       
 a   b       
      
 a  b      
The 2 gradients are parallel.
N1
N2
Say 2 surfaces  a   b   and a  b   intersect at a
curve  . Note that  a   b is a vector normal to the surface
 a   b   and a  b is a vector normal to the surface
a   b    Then  a b  a  b is a vector parallel
to the curve  at the point a   b
Example: Find parametric equations of the tangent line to the
curve of intersection of the paraboloid      and the
ellipsoid         at the point a  b
 a   b       and
a   b         
  a   b   a  b
 a  b  a   b a  b  a  b
â
â
 â
â 
â
â
    â     â      
â
â
 â
â 
Therefore 12, -14, -4 are the direction numbers for the line
which can be reduced to 6, -7, -2. The line is given by





or



                
               