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Let's say we have a point $P (p_x, p_y, p_z)$ and we want to get the position vector to the point. Is there a general way to always get the position vector easliy in $\mathbb{R}^3$? And in $\mathbb{R}^n$?

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On your questions (or any questions for that matter!) it would be nice to upvote any answers you find helpful by clicking on the up arrows above the number on the left side of the answer. More people will chime in with answers if you do so (and continue to accept answers like you have been). Thanks. –  JohnD Jan 2 '13 at 17:03

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Perhaps you overlooked the obvious: If $P$ has coordinates $(x,y,z)$, then the vector whose tail is at the origin and whose tip is at $(x,y,z)$ is simply $x\,i+y\,j+z\,k$ where $i,j,k$ are the standard basis vectors in $\mathbb{R}^3$.

This quickly generalizes to $\mathbb{R}^n$: if $(x_1,x_2,\dots,x_n)$ is a point in $\mathbb{R}^n$, then the vector you seek is simply $x_1\,e_1+x_2\,e_2+\cdots+x_n\,e_n$ where $\{e_1,\dots,e_n\}$ are the standard basis vectors for $\mathbb{R}^n$.

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If I did not misunderstood your question it suffice to take $$\overline{v}=\frac{v}{\langle v, v\rangle}$$ as the position vector for $v\in \mathbb{R}^{3}$.

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You've given a unit vector as the answer, but I do not believe that every position vector is a unit vector. I believe the answer is even simpler. –  mixedmath Dec 27 '12 at 12:04

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