Find the unit vector in the direction of steepest ascent and descent at P? What is the vector that points in a direction of no change?

Question

Consider the function f(x,y)=5x²-5y²-7 and the point (-2, -1).

a) Find the unit vectors that give the direction of the steepest ascent at P?

b) Find the unit vectors that give the direction of the steepest descent at P?

c)Find a vector that points in a direction of no change in the function at P?

Answer 1

The answers to all three questions relate to the gradient of f.

∇f(x,y) = (10x, -10y)

∇f(-2,-1) = (-20, 10)

(a) Steepest ascent is in the direction of the gradient,

normalized into a unit vector: (-2√5/5, √5/5).

(b) Steepest descent is in the direction opposite the gradient,

normalized into a unit vector: (2√5/5, -√5/5).

(c) No change is in direction perpendicular to the above.

There are two normalized unit vectors: (√5/5, 2√5/5), (-√5/5, -2√5/5)

I am sorry, I do not understand why (a) and (b) are asking for the unit vectors in plural. In this case those are unique.