Projectivization of copies of the real tautological line bundle ($\mathbb{P}(k{\gamma }_{\mathbb{R}}^{1,n+1})$)

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 $x+v$.