The book I am using defines a tangent vector to $\mathbb R^3 $ at a point $p$; $\ v_p $ as the line segment $\ p+v $ though both p and v are points in $\mathbb R^3 $. My question is since all points in $ \mathbb R^3 $ can themselves be identified with position vectors, does this mean that every point is tangent vector to $\mathbb R^3 $ at origin and if so is this a specialty of Euclidean spaces due the fact that they come equipped with an origin? Also why is it called the tangent to $\mathbb R^3 $, is $\mathbb R^3$ seen to be embedded in $\mathbb R^4$?
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