  In this paper we study measure theoretical size of the image of naturally
embedded vector bundles in $\mathbb{R}^{n} \times \mathbb{R}^{n}$ under the
codiagonal morphism, i.e. $\Delta_{*}$ in the category of finite dimensional
$\mathbb{R}$-vector spaces. Under very weak smoothness condition we show that
codiagonal of normal bundles always contain an open subset of the ambient
space, and we give corresponding criterions for the tangent bundles. For any
differentiable hypersurface we show that the codiagonal of its tangent bundle
has non-empty interior, unless the hypersurface is contained in a hyperplane.
Assuming further smoothness (e.g. twice differentiable) we show that union of
any family of hyperplanes that covers the hypersurface has maximal possible
Hausdorff dimension. We also define and study a notion of degeneracy of
embedded $C^{1}$ vector bundles over a $C^{1}$ submanifold and show as a
corollary that if the base manifold has at least one non-inflection point then
codiagonal of any $C^{1}$ line bundle over it has positive Lebesgue measure.
Finally we show that codiagonal of any line bundle over an $n$-dimensional
ellipsoid or a convex curve has non-empty interior, and the same assertion also
holds for any non-tangent line bundle over a hyperplane.
