Vector Analysis - Finding perpendicular vectors Given the vector:
$$
\vec{u}=-7\hat{i}-7\hat{j}+7\hat{k}
$$
I need to find another vector $\vec{v} $ that is parallel to the $xy$ plane and perpendicular to the vector $\vec{u} $ .
How can I do it?
I know that any vector that is perpendicular to $\vec{u}$ must satisfy $\vec{u}\cdot \vec{v} =0$ . But how does this help me with finding such a vector?

If you have a vector on the Euclidean plane $\vec{v}=(a,b)$ then the vector $\vec{u}=(-b,a)$ is perpendicular to $\vec{v}.$ We can apply the same idea in $3$-dimensions: 
If we have a vector $\vec{v}=(a,b,c)$ then the following three vectors $$\vec{u}_1=(-b,a,0),\quad \vec{u}_2=(-c,0,a),\quad \vec{u}_3=(0,-c,b),$$ are perpendicular to $\vec{v}.$ One of these gives you the desired vector.
