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## Homework Statement

Find the directional derivative using ##f\left(x,y,z\right)=xy+z^2## at the point (4, 2, 1) in the direction of a vector making an angle of ##\frac{3π}{4}## with ##\nabla f(4, 2, 1)##.

## Homework Equations

##f\left(x,y,z\right)=xy+z^2##

## The Attempt at a Solution

I found the gradient of ##f(4, 2, 1)## as ##\langle 2,4,2 \rangle##.

Now I'm not sure what to do next. I have a 3d vector, but they only give me one angle, so I don't know which orientation the new unit vector is at.

It's taken me about 30 min to type all this up (I keep going back and adding and deleting stuff), and I've got to go to an exam for another class, but I just now thought of this: I think that if I knew which direction I had to rotate I could solve this by finding a vector perpendicular to the gradient vector and a vector opposite of the gradient vector and then adding them together to get a vector at 3π/4 from the gradient vector. Knowing that would allow me to get the unit vector I need to solve this.