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?

  • $\begingroup$ 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. $\endgroup$
    – 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.

Regarding the justification: if we allowed rotations, then any two vectors with the same length would be considered equal, so vectors would not be useful for describing directions. Even if we just allowed rotation by 180 degrees, every vector would be equal to its negative, and this implies that every vector is zero. So, requiring equality of length and direction (including sense) is the only option that makes sense.

I don't know of a name for the geometry arising from studying properties that are invariant under translation only.

  • $\begingroup$ 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. $\endgroup$
    – Shahab
    Aug 10 '14 at 11:12
  • $\begingroup$ You asked 4 questions. I think I have answered 3 of them, now. $\endgroup$
    – bubba
    Aug 10 '14 at 11:55

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