I'm working on a dynamical systems problem and have arrived at a naive question in differential topology? geometric measure theory? 

I have a smooth path $\gamma\colon \mathbb R\to\mathcal G_2(\mathbb R^d)$ (that is $\gamma(t)$ is a 2-dimensional subspace of $\mathbb R^d$ for each $t$). I assume that $\mathbb sR^d$ is equipped with the standard inner product. Now, let $V(t)$ be the collection of vectors in the hypersphere such that for $v\in V(t)$, $v$ is perpendicular to all vectors in $\gamma(t)$. Of course, $V(t)$ is the intersection of the unit sphere with a subspace of $\mathbb R^d$ of dimension $d-2$, so is a set of dimension $d-3$.

I'd like to show that the union of $V(t)$ is of Hausdorff dimension at most $d-2$. 
I already have a kludgy differential equations proof of this, but would like to find a nicer way to do this.

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Can anyone suggest a suitable framework for a nice proof of this kind of result?
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