Finding Directional Derivative

In summary, the conversation discusses finding possible combinations of a and b values that satisfy the equation 3a-4b=4, which represents the direction vector for the given gradient at point P(1,-1). The equation is set equal to the desired directional derivative and it is noted that there are an infinite number of possible combinations for a and b. It is also mentioned that the direction vector should have a magnitude of 1 and a convenient way to write a unit vector making an angle of θ with respect to the positive x-axis is discussed.
  • #1
19
7
Homework Statement
see post
Relevant Equations
∇f⋅u= Direction vector
DDquestion2.PNG
DDQuestion1.PNG


The gradient is < (2x-y), (-x+2y-1) >
at P(1,-1) the gradient is <3, -4>

Since ∇f⋅u= Direction vector, it seems that we should set the equation equal to the desired directional derivative.

< 3, -4 > ⋅ < a, b > = 4

which becomes

3a-4b=4

I thought of making a list of possible combinations of a's and b's which satisfy this equation like so

a, b
corresponding direction vector

0, 1
<0, 1>

(4/3), 0
(for which there is no direction vector??)

2, (1/2)
< 4/√17 , 1/√17 >

But it seems that there are an infinite number of possible combinations. And the question is asking for all of them.

Thank you for your assistance.
 
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  • #2
i am not sure but i don't think it can be infinite in two dimension
i think it is two
##
(3,-4) , (a,b) = 5.1.cos \theta = 4
##
there is two values of theta one negative of the other. maybe in three dimension you might have infinite vectors because they all in a cone of same theta. but in two dimension i think it might be two vectors
 
  • #3
Amadeo said:
Problem Statement: see post
Relevant Equations: ∇f⋅u= Direction vector

View attachment 244655View attachment 244656

The gradient is < (2x-y), (-x+2y-1) >
at P(1,-1) the gradient is <3, -4>

Since ∇f⋅u= Direction vector, it seems that we should set the equation equal to the desired directional derivative.

< 3, -4 > ⋅ < a, b > = 4

which becomes

3a-4b=4

I thought of making a list of possible combinations of a's and b's which satisfy this equation like so

a, b
corresponding direction vector

0, 1
<0, 1>

(4/3), 0
(for which there is no direction vector??)

2, (1/2)
< 4/√17 , 1/√17 >

But it seems that there are an infinite number of possible combinations. And the question is asking for all of them.

Thank you for your assistance.
The direction vector, ##\vec{u}##, should have a magnitude of 1, i.e, it should be a unit vector.

What is a convenient way to write a unit vector, in particular a unit vector making a angle of ##\theta## with respect to the positive x-axis ?
 

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