Thread: Gradient vector question

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>=√2²+3²
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

So what does it mean for two vectors to be in the same direction?

I think I got it, <a,b>=<2,3> / √13 = <2/√13, 3/√13>...I can't believe i didn't realise that, i was trying all sorts. Looks like the answer is right, since I checked its magnitude, and it's 1. Looks like another wrong answer in the book. :?

Hey hey hey check to see if they're equal!

Oh yeh, 1/√13 = √13/13, so dividing both by √13 gives:

1/√13√13 = 1/13
1/13=1/13

I don't know why they used √13/13 though, when surely 1/√13 is much more straigtforward, since a unit vector is just the vector divided by its magnitude, √13.

Somebody came up with the convention once that the denominator should always be rationalized. I bet it was developed by or at the behest of a teacher who was sick of having to check whether various expressions were equal. And, now that I think about this, I'm once again frightened that a lot of math teachers aren't that good at math.

You're right, 1/sqrt(13) is a lot more brief than sqrt(13)/13, and it displays the algebraic property of the number--namely that its square is 1/13--more clearly.

In general, you should make an effort to become fluent in manipulating expressions. It'll be useful for checking your answers, and it's a good strategy for solving a lot of problems.

Thanks for the help. I just need to keep doing problems over and over until I get used to it. I always find with maths, that if I'm not actively solving problems, then within the space of a year or so, I'll be extremely rusty. That has morealess been the case with me before I started reading this book....I finished calculus, did almost no challenging maths for a year, and then started this book. I also hear that universities/companies don't like maths students/employees to be, to take gap years because of this reason....I heard that was the case over here anyway.

Well, then, make sure you're always doing some math! We're always here to support you in such endeavors.

Thanks. Actually I feel quite honoured to get help from a Phd maths student!