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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