Finding a Directional Derivative Given Other Directional Derivatives

[Amrator]

Homework Statement

Suppose $D_if(P) = 2$ and $D_jf(P) = -1$. Also suppose that $D_uf(P) = 2 \sqrt{3}$ when $u = 3^{-1/2} \hat i + 3^{-1/2} \hat j + 3^{-1/2} \hat k$. Find $D_vf(P)$ where $v = 3^{-1/2}(\hat i + \hat j - \hat k)$.

Homework Equations

The Attempt at a Solution

$$2\sqrt{3} = ∇f ⋅ 3^{-1/2}(\hat i + \hat j - \hat k)$$
$$= 3^{-1/2}(∂f/∂x + ∂f/∂y + ∂f/∂z)$$
$$6 = (∂f/∂x + ∂f/∂y + ∂f/∂z)$$

This is where I'm stuck. I would appreciate hints.

 

[Ray Vickson]

Homework Statement

Suppose $D_if(P) = 2$ and $D_jf(P) = -1$. Also suppose that $D_uf(P) = 2 \sqrt{3}$ when $u = 3^{-1/2} \hat i + 3^{-1/2} \hat j + 3^{-1/2} \hat k$. Find $D_vf(P)$ where $v = 3^{-1/2}(\hat i + \hat j - \hat k)$.

Homework Equations

The Attempt at a Solution

$$2\sqrt{3} = ∇f ⋅ 3^{-1/2}(\hat i + \hat j - \hat k)$$
$$= 3^{-1/2}(∂f/∂x + ∂f/∂y + ∂f/∂z)$$
$$6 = (∂f/∂x + ∂f/∂y + ∂f/∂z)$$

This is where I'm stuck. I would appreciate hints.

$v$ is a linear combination of $i, j$ and $u$.

 

[Amrator]

$v = 1/\sqrt{3} <1, 0, 0> + 1/\sqrt{3} <0, 1, 0> - 1/\sqrt{3} <1/\sqrt{3}, 1/\sqrt{3}, 1/\sqrt{3}>$
$f(x,y,z) = 1/\sqrt{3} x + g(y,z)$
$f(x,y,z) = 1/\sqrt{3} y + h(x,z)$

I'm not sure what ∂f/∂z would be.