Tangent space in the book "Differential Forms and Applications"

Question

In the book "Differential Forms and Applications", the author defines the tangent space of $\mathbb{R}^{3}$ at $p$ ($p \in \mathbb{R}^{3}$) as $\mathbb{R}^{3}_{p}=\{q-p; q \in \mathbb{R}^{3}\}$. My question is, this definition coincides with the definition which the tangent space of $\mathbb{R}^{3}$ at $p$ is $\mathbb{R}^{3}_{p}=\varphi'(p).\mathbb{R}^{3}$, where $\varphi=Id_{\mathbb{R}^{3}}$? Why the author adopts the first definiton?