# Why are two vectors that are parallel equivalent?

Why are two parallel vectors with the same magnitude equivalent?

Why is their start point irrelevant?

How can a vector starting at $\,(0, -10)\,$ going to $\,(10, 0)\,$ be the same as a vector starting at $\,(10, 10)\,$ and going to $\,(20, 20)\,$?

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The start point certainly can be relevant. Imagine I choose to drop a rock either onto your foot or two feet away onto the floor. I suspect the start point for that force vector (both pointing down, with the same magnitude) will be important to you. – KCd Aug 27 '12 at 1:39
What is a vector? The answer to the question that’s posed must depend on your definition of “vector”. For me, a vector is not an arrow but something rather more abstract that is only described by an arrow. Others have different conceptions. – Lubin Aug 27 '12 at 3:56
@Lubin can you elaborate on that? That's kind of what my question was asking in a weird way. What is a vector? – mr real lyfe Aug 27 '12 at 3:58

Vectors denote a direction and length, rather than start and end points. Sometimes they are called free vectors. You can translate the start and end points of a vector by the same amount, and still get the same vector. To see why observe that for each coordinate we have $x_1-x_2 = (x_1+\Delta)-(x_2+\Delta)$. So two vectors with same direction & length are indistinguishable in that sense.
So two parallel vectors have the same direction. The direction of a vector $v = (x ,y)$ is given by the unit vector $u = (\dfrac{x}{\sqrt{x^2 + y^2}}, \dfrac{y}{\sqrt{x^2 + y^2}})$ of length $1$. The example is 2D for simplicity.