The question is: "For the function f(x,y) = 2x+3y find a unit vector that points in the direction of the maximum rate of change at the point (1,1)"

first of all, I found the gradient vector, ∇f = <2,3>, and this should be the same for any co-ordinates. Then I used this formula:

V.∇f = |V||∇f|cosθ

if |V| = 1, and cosθ = 1 (same direction as gradient vector), then

V.∇f=|∇f|

<a,b>.<2,3>=√2<sup>2</sup>+3<sup>2</sup>

2a+3b = √13

Now I'm lost on how to solve for a and b. If I square the expression on the left, and re-arrange, I just get a polynomial with two variables, which i can't solve.

The answer the book gives is:

<(2√13)/13, (3√13)/13>

Thanks