I am trying to derive the equation of a tangent plane at some point [tex](x_0, y_0)[/tex] on a surface using vectors. This is how I have been trying to do it: The tangent line at [tex](x_0, y_0)[/tex] in the x-direction is [tex]z=z_0+f_x(x-x_0)[/tex] so the vector parallel to it is [tex]L_1=<(x-x_0), 0, (z-z_0)>[/tex]. Similarly, the vector parallel to the tangent line with respect to y is [tex]<(0, (y-y_0), (z-z_0)>[/tex]. Taking the cross product, I got the normal vector [tex]<-(z-z_0)(y-y_0), -(x-x_0)(z-z_0), (x-x_0)(y-y_0)>[/tex] Then taking the dot product between the normal vector and a vector in the plane ([tex]L_2-L_1[/tex]), I got a formula which does equal zero but from which I cannot seem to derive the desired equation. This is what I keep getting: [tex](x-x_0)(y-y_0)(z-z_0)-(x-x_0)(y-y_0)(z-z_0)=0[/tex] I've tried different vector representations of the lines but I keep getting the same result.