I don't seem to understand how to approach this type of question:
For the following function, find the direction in which the directional derivative is zero:

w = xy + yz + xz, at (1,-1,2)

Question

I don't seem to understand how to approach this type of question:
For the following function, find the direction in which the directional derivative is zero:

w = xy + yz + xz, at (1,-1,2)

anonymous · Answer

I calculated the gradient at this point: $$grad(w) = <1,3,0>$$ and now i think that I'm supposed to solve $$*<1,3,0> = 0$$ and: $$x^2+y^2+z^2 = 1$$ but I'm lacking a 3rd equation for z... in the answer they just set z = c (for all c) and get some vector but it differs from mine and I don't get how they came to it... (my grad is correct though).

anonymous · Answer

Having multiple solutions is okay.

anonymous · Answer

Or are they expecting only one solution?

anonymous · Answer

that's their answer

anonymous · Answer

the top is -3i+j+ck

anonymous · Answer

Yes, there are infinite solutions, that why they have $c$ as a free variable I suppose?

anonymous · Answer

yep, but I don't see how they came to this expression

anonymous · Answer

Well... $$
x+3y = 0
$$

anteater · Answer

What is their answer?

anonymous · Answer

Such is how they get the $-3,1$

anonymous · Answer

$z$ becomes a free variable.

anonymous · Answer

However, I got this funky feeling that there may be more solutions than just that...

anonymous · Answer

Wait, never mind.  They've got all the solutions.

anonymous · Answer

Well, thank you very much. I will play a bit more with the equations to see what can I get out of them. As far as I can see it's just algebric manipulation from now.