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 >#