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Answer 1 · 2017-04-30T21:39:15

See below.

The normal vector to a surface #f(x,y,z)=0# is given by

#grad f=(f_x,f_y,f_z)#

Given

#p_0 = (1,2,4)# and
#p = (x,y,z)#

the tangent plane at this point is given by

#Pi-> << p - p_0, vec n >># where

#<< cdot, cdot >># represents the scalar product of two vectors and

#vec n = grad f(p_0) = (2x_0,2y_0,1)=(2,4,4)# so

#(x-1)2+(y-2)4+(z-4)4=0#

The normal line is given by

#L->p = p_0 + lambda vec n# or

#{(x=1+2lambda),(y=2+4lambda),(z=4+4lambda):}#

Answer 2 · 2017-04-30T21:59:58

We are using the directional derivative which tells us that the normal vector is the gradient, ie :

#mathbf n = mathbf nabla f(mathbf x) = < 2x, 2y, 1 >_{(1,2,4)} = < 2, 4, 1 >#

So, the tangent plane has #mathbf n# as it's normal vector, and also, like any other plane, has equation:

#(mathbf r - mathbf r_0) cdot mathbf n = 0 implies mathbf rcdot mathbf n = color(blue)(mathbf r_0 cdot mathbf n) = color(blue)(alpha)#

The normal line passes through point #(1,2,4)#, so #mathbfr_o = < 1,2,4 > # and #mathbf r_0 cdot mathbf n = <1,2,4> cdot < 2, 4, 1> = 14 = alpha#

So the tangent plane is:

#2x + 4y + z = 14#

The normal line, #mathbf l#, passes through #(1,2,4)# and has direction # < 2, 4, 1 >#. With #lambda# as the parameter:

#mathbfl = <1,2,4> + lambda < 2, 4, 1 >#