Projectivization of a real vector bundle ($\mathbb{P}(E)$)

The projectivization $\mathbb{P}(E)\to B$ has fiber over $b$ equal to the space of one-dimensional linear subspaces of $E_b$.

Let $p:\mathbb{P}(E)\to M$, let $L_E$ be the tautological line, and let ${\omega }_E$ be its orthogonal complement in $p^{\ast }E$. ThenThe second summand is the vertical tangent bundle.

Tensoring a fixed line with ${\mathbb{R}}^k$ does not change its projective directions, soIf $x$ and $v$ are the degree-one generators from the two factors, then the tautological line over this projective bundle has first Stiefel–Whitney class $x+v$.