How Do You Calculate a Perpendicular Unit Vector in Multivariable Functions?

[Calpalned]

Homework Statement

Suppose $F(x, y, z, u, v) = xy^2 + yz^2 + zu^2 + uv^2 + vx^2$ Standing at the point $(1, 1, 1, 1, 1)$ imagine moving in a direction $\vec w$ where $\vec w$ is a unit vector. Find the components of a vector $\vec u$ such that $D_\vec u F = 0$
Remember $\vec w$ needs to be unit vector.

Homework Equations

Directional vector = $\nabla F \cdotp \vec w$

The Attempt at a Solution

Directional vector = $<F_x, F_y, F_z, F_u, F_v> \cdotp \vec w = 0$
So $\nabla F$ and $\vec w$ are perpendicular.
If $\vec w = <a, b,c, d, e>$ then the components of $\vec w$ must satisfy $aF_x + bF_y + cF_z + dF_u + eF_v = 0$
To sum it up $\vec w = <a, b,c, d, e> \frac {1}{|w|} =$ unit vector

I only got 4/10 points on this question, so I made a mistake somewhere...

 

[Brian T]

Is that all of the work you did? You have everything in implicit form...
Your vector <a,b,c,d,e> could be any five numbers at this point.

You need to go more specific