Multivariable calculus yummy in my tummy

[mathwiz123]

[Question]
Hi, my teacher gave us this problem, and he couldn't figure out why
method was incorrect and why I got the answer I did.

Given we know the gradient slope = <-56,1.886> at the point (2,0) on a
surface f(x,y), in what direction, expressed as a unit vector, is f
increasing most rapidly?





[Difficulty]
I solved the problem like this:

Max slope = magnitude of gradient slope
(gradient slope) dot (unit vector) = 56.03
-56.03Ux + 1.866Uy = 56.03
sqt(Ux^2 + Uy^2) = 1

Solving the system Ux= -.9977 Uy=.0672

The only problem is, by definition, I should be able to get the unit
vector by taking the gradient slope vector and dividing by the
magnitude of the gradient vector. or,

<-56/56.03, 1.886/56.03> = <Ux,Uy> = <-.9994,.0336>

The weird thing is, the correct Uy value is almost exactly half of
mine...what's going on?

[Thoughts]

I tried a similar technique for finding where the ∇f = 0

<-56, 1.866> dot (unit vector) = 0

-56.03Ux + 1.886Uy = 0
sqt(Ux^2 + Uy^2) = 1

Solving the system this time I got the correct answer, why here and
not there?

 

[HallsofIvy]

[Question]
Hi, my teacher gave us this problem, and he couldn't figure out why
method was incorrect and why I got the answer I did.

Given we know the gradient slope = <-56,1.886> at the point (2,0) on a
surface f(x,y), in what direction, expressed as a unit vector, is f
increasing most rapidly?





[Difficulty]
I solved the problem like this:

Max slope = magnitude of gradient slope
(gradient slope) dot (unit vector) = 56.03
-56.03Ux + 1.866Uy = 56.03
sqt(Ux^2 + Uy^2) = 1

Solving the system Ux= -.9977 Uy=.0672

I understand that the 56.03 is the magnitude of the given vector but the coefficient of Ux should be -56, not -56.03.

The only problem is, by definition, I should be able to get the unit
vector by taking the gradient slope vector and dividing by the
magnitude of the gradient vector. or,

<-56/56.03, 1.886/56.03> = <Ux,Uy> = <-.9994,.0336>

The weird thing is, the correct Uy value is almost exactly half of
mine...what's going on?

[Thoughts]

I tried a similar technique for finding where the ∇f = 0

<-56, 1.866> dot (unit vector) = 0

-56.03Ux + 1.886Uy = 0
sqt(Ux^2 + Uy^2) = 1

Solving the system this time I got the correct answer, why here and
not there?