Find equations for the (a) Tangent plane and (b) Normal line at the point P 0 on the given surface x 2 y 2 2xy x 3y z 4, P 0 (2, 3, 18) ...

a To find the equation of the tangent plane to the surface at the point P02 3 18 we need to find the normal vector to the surface at that point The normal vector is given by the gradient vector of the surface evaluated at P0 fx y z 2x 2y 1 2y 2x 3 1 f2 3 18 1 7 1 Thus the equation of the tangent plane at P0 is x 7y z 34 b To find the equation of the normal line to the surface at P0 we can use the fact that the normal line is perpendicular to the tangent plane Thus the direction vector of the normal line is the same as the normal vector of the tangent plane which is 1 7 1 We can use this direction vector and the point P0 ...