How to convert the vector equation of a plane into the parametric form?

Question

If we know the normal vector of the plane, can we take the cross product of this vector and any other vector ( e.g, 0,0,1 ), and then do the same thing ( n x 0,1,0 ) for example. Both of the resultant vectors are perpendicular to the normal, and hence lie in the plane. If they turn out to be linearly independent, can we use them to write out the equation of our plane in parametric form?

Answer 1

Yes, this works (mostly; see below). To see this, let $\mathbf{n}$ be a normal vector and $\mathbf{v}$ be arbitrary. Then the vectors $\mathbf{n}$, $\mathbf{v}\times\mathbf{n}$, and $\mathbf{n}\times(\mathbf{v}\times\mathbf{n})$ are mutually orthogonal. They're also linearly independent, so no need to worry about that. Your plane can then be parameterized as $$ \mathbf{P}(s,t) = (\mathbf{v}\times\mathbf{n})s + [\mathbf{n}\times(\mathbf{v}\times\mathbf{n})]t + \mathbf{x} $$ where $\mathbf{x}$ is any point on the plane.

However, this doesn't work if $\mathbf{v}\times\mathbf{n} = 0$. This can obviously be avoided by judicious choice of $\mathbf{v}$, but it's something to be careful of.