Let $\mathbb{F}_{q}$ be a finite field of characteristic 2 and$O_2(\mathbb{F}_{q})^{+}$ be the 2-dimensional orthogonal group of plus typeover $\mathbb{F}_{q}$. Consider the standard representation $V$ of$O_2(\mathbb{F}_{q})^{+}$ and the ring of vector invariants$\mathbb{F}_{q}[mV]^{O_2(\mathbb{F}_{q})^{+}}$ for any $m\in \mathbb{N}^{+}$.We prove a first main theorem for $(O_2(\mathbb{F}_{q})^{+},V)$, i.e., we finda minimal generating set for $\mathbb{F}_{q}[mV]^{O_2(\mathbb{F}_{q})^{+}}$. Asa consequence, we derive the Noether number$\beta_{mV}(O_2(\mathbb{F}_{q})^{+})=\max\{q-1,m\}$. We construct a free basisfor $\mathbb{F}_{q}[2V]^{O_2(\mathbb{F}_{q})^{+}}$ over a suitably chosenhomogeneous system of parameters. We also obtain a generating set of theHilbert ideal for $\mathbb{F}_{q}[mV]^{O_2(\mathbb{F}_{q})^{+}}$ which showsthat the Hilbert ideal can be generated by invariants of degree $\leqslantq-1=\frac{|O_2(\mathbb{F}_{q})^{+}|}{2}$, confirming Derksen-Kemper'sconjecture \cite[Conjecture 3.8.6 (b)]{DK2002} in this particular case.
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