Let $Q \cong \mathbb P^1$ be a quadric curve in $\mathbb P^2$. Consider the following rank two vector bundle $V$ on $Q$: the fiber of $V$ over a point $p$ of $Q$ is the two-dimensional subspace $V_p$ of $\mathbb A^3$ such that the projectivization $\mathbb P(V_p)$ is the tangent line to $Q$ at $p$ inside $\mathbb P^2 = \mathbb P(\mathbb A^3)$. Question: what are $a$ and $b$ such that $V \cong \mathcal O(a) \oplus \mathcal O(b)$ as a rank 2 vector bundle on $Q \cong \mathbb P^1$?
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In other words, you want to understand the extension $$ 0 \to \mathcal{O}_Q \to V \to \mathcal{T}_Q \to 0 $$ induced by the restriction to $Q$ of the canonical extension $$ 0 \to \mathcal{O}_{\mathbb{P}^2} \to \mathcal{O}_{\mathbb{P}^2}(1)^{\oplus 3} \to \mathcal{T}_{\mathbb{P}^2} \to 0 $$ with respect to the natural embedding $\mathcal{T}_{Q} \to \mathcal{T}_{\mathbb{P}^2}\vert_Q$. For this note that the normal exact sequence of $Q$ induces an exact sequence $$ 0 \to V \to \mathcal{O}_{\mathbb{P}^2}(1)\vert_Q^{\oplus 3} \to \mathcal{N}_{Q/\mathbb{P}^2} \to 0, $$ which can be explicitly rewritten as $$ 0 \to V \to \mathcal{O}_{Q}(2)^{\oplus 3} \to \mathcal{O}_{Q}(4) \to 0. $$ Its second arrow is given by the restrictions to $Q$ of the partial derivatives of the equation of $Q$. If you choose coordinates such that the equation of $Q$ is $xz - y^2$, the derivatives are $(z,-2y,x)$, their restrictions to $Q$ are $(v^2,-2uv,u^2)$ in appropriate coordinates, hence $$ V \cong \mathcal{O}_{Q}(1)^{\oplus 2}. $$
