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 $\,(20, 20)\,$?
 A: 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.
A: This is because both these vectors have the same magnitude and direction. Both may have different frames of reference, but for all computational purposes, they are identical. We have centered each vector by taking the starting point as the origin and retaining only the displacement with respect to that point.
However, if for examples, these vectors signify forces acting on a rigid (non-point mass) body, then they could result in different torques and hence have different meaning.
A: Components of a vector determine the length and the direction of the vector, but not it's basepoint. Therefore 2 vectors are equivalent if and only if they have the same components, but this gives the idea that any 2 parallel vectors are equal though they have different base points. To avoid this confusion, we take that all vectors are based on the origin unless stated.
