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I am thinking specifically about $\mathbb{R}^2$ so I can visualize things.

By "origin" I mean that they start at the same point.

When we graphically represnt vectors we don't care where the starting point is (i.e. where the vector begins does not affect the vector; the vector $(1,2)^T$ is the same whether we draw it at the origin of $(10,10)$)

Since the point where we draw a vector starting from doesn't matter, Then I should be able to draw two different vectors that start at two different points.

Normally, if I want to graphically represent the angle between these two vectors, I would reposition them so that they start at the same point.Why is this the correct way to represent the angle between two vectors? (or maybe it isn't?)

That is, is there a part of the definition of the angle between two vectors that suggests that, graphically, we should represent them as originating from the same point?

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  • $\begingroup$ In math a vector is $v \in \mathbb{R}^2$ is defined to be just an ordered pair $(x,y)$. You can think of it as emanating from some other point if you want, but that is not part of the definition of the vector. So it would not make sense to give a definition of an angle between two vectors that was different depending on where you place them. Otherwise it would not be the angle between two vectors, but the angle between some other objects (two vectors plus their starting points). $\endgroup$
    – Jair Taylor
    Aug 31, 2018 at 0:45
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    $\begingroup$ Extend your arbitrarily-positioned vectors into lines and measure the angle at their intersection. $\endgroup$
    – amd
    Aug 31, 2018 at 0:59

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Define vectors as directed line segments with equality between two vectors holding iff they point in the same direction and have the same length. You can now define the angle without insisting that the two vectors have the same origin.

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Actually, if you just position the line segments that represent your vectors somewhere such that they intersect, the geometric angle between those line segments where they cross will be the angle between the vectors.

Making the arrows cross right at their origin end is just one extreme of this.

(In general there are two "vertical" angles where the line segments cross; you need to take the direction of the vectors into account to find out which of those is the angle between the vectors).

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As you know vectors do not depend on their initial points so it is helpful if we consider the angle between two vectors where they are in standard position which means they start at the origin.

That is particularly useful when finding the angles using the dot product.

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The angle between two as you are right is looked by joining their tails.The angle so formed is the one which manifests itself in the dot product formula.

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