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In this paper we study the following type of functions $f:\mathcal{Q}_{\mathbb{R}_{3}} \to \mathbb{R}_{3}$, where $\mathcal{Q}_{\mathbb{R}_3}$ is the quadratic cone of the algebra$\mathbb{R}_{3}$. From the fact that it is possible to write the algebra $\mathbb{R}_{3}$ as a direct sum of quaternions, we get the observation that itis possible to find a clever representation for $ \mathcal{Q}_{\mathbb{R}_3}$.By using this result a slice regular theory was introduced and a Cauchy formulais discussed. Moreover, a detailed study of the zeros is performed. Finally, wefind a formula for the determinant of a matrix with entries in $\mathcal{Q}_{\mathbb{R}_3}$.

On the Quadratic Cone of $\mathbb{R}_3.$