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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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I like visualizing $\mathbb{R}^2$ better. Tangent vectors of elements in the plane are intuitively in the plane. So it should be no surprise that they're in the plane. It's when we start having manifolds that the tangential aspect comes into bigger play. – mixedmath Apr 16 '12 at 16:59
What's special about $\mathbf R^n$ is that all of the tangent spaces are canonically identified. On a general manifold there is no way of doing this, but see the Wikipedia article on connections. – Dylan Moreland Apr 16 '12 at 17:08

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