ࡱ> =?<zg 9`bjbjVV 8fr<r<%<<,,,8dt,;t ",,737373:::::::${<-?h:]3.7333:<<,=;2929293R<V,p:293:292929  4~29:S;0;29?5?29?29t73K329Y3 e3S737373::Z7737373;3333?737373737373737373 : Section 11.6: Tangent Planes and Normal Lines Practice HW from Larson Textbook (not to hand in) p. 710 # 3-25 odd Normal Lines to Surfaces Recall that z = f (x, y) gives a 3D surface in space. We want to form the following functions of 3 variables  EMBED Equation.3  Note that the function  EMBED Equation.3  is obtained by moving all terms to one side of an equation and setting them equal to zero. We use the following basic fact. Fact: Given a point  EMBED Equation.3  on a surface, the gradient of F at this point  EMBED Equation.3  is a vector orthogonal (normal) to the surface  EMBED Equation.3 .         Example 1: Find a unit normal vector to the surface  EMBED Equation.3  at the point (2, 1, 2) Solution: % Tangent Planes Using the gradient, we can find a equation of a plane tangent to a surface and a line normal to a surface. Consider the following:      Recall that to write equation of a plane, we need a point on the plane and a normal vector. Since  EMBED Equation.3  represents a normal vector to the surface (and the tangent plane), its components can be used to write the equation of the tangent plane at the point  EMBED Equation.3 . The equation of the tangent plane is given as follows:  EMBED Equation.3 . Recall, to write the equation of a line in 3D space, we need a point and a parallel vector. Since  EMBED Equation.3  is a vector normal to the surface, it would be parallel to any line normal to the surface at  EMBED Equation.3 . Thus, the parametric equations of the normal line are:  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3  We summarize these results as follows.  Tangent Plane and Normal Line Equations to a Surface Given a surface z = f (x, y) in 3D, form the function  EMBED Equation.3 of three variables. Then the equation of the tangent plane to the surface z = f (x, y) at the point  EMBED Equation.3  is given by  EMBED Equation.3 . The parametric equations of the normal line through the point  EMBED Equation.3  are given by  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3  Note: Recall that to find the symmetric equations of a line, take the parametric equations, solve for t, and set the results equal. Example 2: Find the equation of the tangent plane and the parametric and symmetric equations for the normal line to the surface  EMBED Equation.3  at the point (2, 1, 2). Solution: % Note: The following graph using Maple shows the graph of the sphere  EMBED Equation.3  with the tangent plane and normal line at the point (2, 1, 2).  Example 3: Find the equation of the tangent plane and the parametric and symmetric equations for the normal line to the surface  EMBED Equation.3  at the point  EMBED Equation.3 . Solution: We start by setting  EMBED Equation.3  and computing the function of 3 variables  EMBED Equation.3  Recall that to get an equation of any plane, including a tangent plane, we need a point and a normal vector. We are given the point  EMBED Equation.3 . The normal vector comes from computing the gradient vector of F at this point. Recall that for a given point  EMBED Equation.3 , the gradient vector at this point is given by the formula  EMBED Equation.3  Computing the necessary partial derivatives, we obtain  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  The given point is  EMBED Equation.3 . Thus, since  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3 , the gradient vector of F at the point  EMBED Equation.3  is  EMBED Equation.3  We use the components of the gradient vector to write the equation of the tangent plane using the formula (continued on next page)  EMBED Equation.3  At the point  EMBED Equation.3 , this formula becomes  EMBED Equation.3  Using the calculations for the partial derivatives given on the previous page, this equation becomes  EMBED Equation.3  or  EMBED Equation.3  We can expand this equation to get it in general form. Doing this gives  EMBED Equation.3  and when combining like terms, we have the equation of the tangent plane   EMBED Equation.3 . The parametric equations of the normal line through the point  EMBED Equation.3  are given by  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3  Using the calculations we computed above where that  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3 , we obtain  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3  which, when simplified, gives (continued on next page)  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3  If we want to convert this these equations to symmetric form, we can take the last two equations of the previous result and solve for t. This gives  EMBED Equation.3  and  EMBED Equation.3 . Equation gives the symmetric equations of the normal line.   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