Comparing a vector with a directed line segment

Let $x$ and $y$ be two vectors in $\mathbb{R}^2$. The parallelogram law of vector addition says that the vector corresponding to the diagonal of the parallelogram formed by these vectors is $x+y$. For the same parallelogram consider the other diagonal directed from $x$ to $y$. Call this diagonal $\vec{D}$.

The vector $y-x$ can be identified with $\vec{D}$ since we see geometrically that it has the same length and the same slope. This is also intuitively obvious because a rigid motion (translation) can superimpose $y-x$ on $\vec{D}$.

However we may identify $\vec{D}$ with $x-y$ as well since they too have the same slope and length. Moreover a rotation and a translation can superimpose $x-y$ on $\vec{D}$ as well.

Yet it is geometrically obvious that $x-y$ is the opposite direction of $\vec{D}$.

Can someone explain why we should identify $\vec{D}$ with $y-x$ only? Is it correct to say that we should identify a vector with a directed line segment if the two are related by a translation only? If so, what is the justification behind this and is there a name for the geometry arising from studying properties invariant under translation only?

• You have declared "$\overrightarrow{D}$" as the name of the "diagonal directed from $x$ to $y$". That is why $\overrightarrow{D}$ is associated with $y-x$; your definition uses the specific direction. The "diagonal directed from $y$ to $x$" would be $x-y$; you could have called that "$\overrightarrow{D}$" if you'd wanted, but you can't use "$\overrightarrow{D}$" for both directions.
– Blue
Aug 10 '14 at 12:00

Vectors in $\mathbb{R}^2$ or $\mathbb{R}^3$ can be regarded (intuitively) as directed line segments. The word "directed" implies that direction/sense is important. The directed line segment $\vec{PQ}$ is not the same as the directed line segment $\vec{QP}$.
The vectors $\mathbf{x} - \mathbf{y}$ and $\mathbf{y} - \mathbf{x}$ are not the same since they don't have the same direction. They have the same length, and they are parallel (some people would say "anti-parallel"), but they do not have the same sense.
• Thanks for answering, but my question is still left unanswered. I perfectly understand that $x-y$ and $y-x$ are not the same since they have opposite directions. Please read the last two sentences again. Aug 10 '14 at 11:12