Solve this Directional Derivative question

Question

" You are given the following information about a differentiable function f(x, y, z):

1)At Po, f increases in the direction of the vector A=i+2j+2k at a rate of 2.

2)At Po. f decreases in the direction of the vector B = 2i+ 3j+6k at a rate of 4

Using only this information and knowing nothing else about the function f, find a nonzero vector such that the directional derivative of f at Po in the direction of C is 0. "

Here is what I have done so far.

I put df/dx(Po) = a, df/dy(Po) = b, df/dz(Po) = c (all are partial derivatives, and remembering that directional derivative means $\nabla f(P_0)$ dot the unit vector).

Using the above 2 conditions, I got:

$$a + 2b + 2c = 2\sqrt{1^2+2^2+2^2}$$

$$2a + 3b + 6c = 4\sqrt{2^2+3^2+6^2}$$

if X=[x,y,z] is our required vector, we need to solve ax + by + cz = 0

So how do I solve this?

Answer 1

Hint:

Solve for $b,c$ in terms of $a$:

• $2b=2-2c-a=2-2(\frac{4-2a-3b}{6})-a\iff b=\frac{2}{3}-\frac{a}{3}$
• $2c=2-2b-a=2-2(\frac{4-2a-6c}{3})-a\iff c=\frac{1}{3}-\frac{a}{6}$

Can you take it from here?