Let $\mathcal F$ be a locally free $\mathcal O_X$-module. Then $\mathcal R := \bigoplus_n \mathcal F^{\otimes n}$, the tensor product being over $\mathcal O_X$, is a sheaf of rings, and we can take its $\bf Spec$ to get a space over $X$. That space is the corresponding vector bundle.

$\mathcal R$'s grading is what gives the dilation action on the fibers. The map $\mathcal F \to (\mathcal F \otimes \mathcal O_X) \oplus (\mathcal O_X \otimes \mathcal F)$, $f \mapsto (f\otimes 1) + (1\otimes f)$ induces a cocommutative comultiplication $\mathcal R \to \mathcal R \otimes \mathcal R$, which gives the vector addition on ${\bf Spec}\ \mathcal R$, I think.

Let $\mathcal F$ be a locally free $\mathcal O_X$-module. Then $\mathcal R := Sym_{\mathcal O_X}(\mathcal F)$, the tensor products being over $\mathcal O_X$, is a sheaf of rings, and we can take its $\bf Spec$ to get a space over $X$. That space is the corresponding vector bundle.

$\mathcal R$'s grading is what gives the dilation action on the fibers. The map $\mathcal F \to (\mathcal F \otimes \mathcal O_X) \oplus (\mathcal O_X \otimes \mathcal F)$, $f \mapsto (f\otimes 1) + (1\otimes f)$ induces a cocommutative comultiplication $\mathcal R \to \mathcal R \otimes \mathcal R$, which gives the vector addition on ${\bf Spec}\ \mathcal R$, I think.