MHB What is the definition of dividing vectors in R^2 using points P, Q, R, and S?

[Poirot1]

If P,Q,R,S are 4 points in R^2 then we have the lines PQ and RS. How do we define $\frac{PQ}{RS}$ ?

 

[chisigma]

If P,Q,R,S are 4 points in R^2 then we have the lines PQ and RS. How do we define $\frac{PQ}{RS}$ ?

It is well known that there are two different vector products: the scalar product and the vectorial product. If we hypothise to define a sort of 'inverse vector' we must distinguish between 'scalar inverse' and 'vectorial inverse'. The scalar inverse seems to be reasonably comfortable to manage... the vectorial inverse probably is a little problematic to manage...Kind regards $\chi$ $\sigma$

 

[chisigma]

The dot product of two vectors X and Y is defined as...

$\displaystyle X \cdot Y = |X|\ |Y|\ \cos \theta$ (1)

... where $\theta$ is the angle between vectors and |*| is the norm. Let's suppose that the 'scalar inverse' of a vector X can be defined as a vector $X^{-1}$ so that is...

$\displaystyle X \cdot X^{-1} = 1$ (2)

It is almost immediate that $X^{-1}$ in that case is not univocally defined because different combinations of $|X^{-1}|$ and $\theta$ can satisfy (2)... it seems that we are not on the right way! (Malthe)...

Kind regards

$\chi$ $\sigma$

 

[Poirot1]

look up ceva's theorem on wikipedia. That's what I'm on about.

 

[Evgeny.Makarov]

look up ceva's theorem on wikipedia. That's what I'm on about.

Wikipedia says the following about Menelaus' theorem, which is similar.

This equation uses signed lengths of segments, in other words the length AB is taken to be positive or negative according to whether A is to the left or right of B in some fixed orientation of the line. For example, AF/FB is defined as having positive value when F is between A and B and negative otherwise.

So, it involves ratios of signed segment lengths.

 

[Ackbach]

If P,Q,R,S are 4 points in R^2 then we have the lines PQ and RS. How do we define $\frac{PQ}{RS}$ ?

Maybe I'm missing something here, but it looks to me like $PQ$ is the length of the vector from point $P$ to point $Q$, and $RS$ is the length of the vector from point $R$ to point $S$. Then the fraction $(PQ)/(RS)$ is normal real division: one length divided by another. $PQ$ is a common, though not, to my mind, the best, notation for the length of that line segment.

Is there more context to the problem?

 

[Evgeny.Makarov]

In Menelaus' theorem, the product of ratios is -1, so these are directed lengths, not simple lengths.

 

[Poirot1]

Yes this is a simple misunderstanding of notation.

 

[Ackbach]

Yes this is a simple misunderstanding of notation.

If you mean I'm misunderstanding the notation, then I would claim the notation is atrocious. I know for a fact that I have seen this notation used for the simple length of a line segment.

Perhaps you could use something like $\overset{\pm}{\overline{PQ}}$?

 

[Poirot1]

You are misunderstanding now, I was referring to myself.