Talk:Direction (geometry)

relative direction

@Fgnievinski, I tried to look up "relative direction", and couldn't find sources describing it the way you had here, so I took that definition out. The sources I could find mentioning relative direction seemed to either use it to mean the same as what we're calling "direction" here, or else meant direction relative to some changing reference frame (compared to absolute direction relative to some fixed reference frame). Do you have any sources with the "vector from a given point" meaning, either as a definition or even in practical use?

I'll see if I can find some better sources defining "direction" in general; unfortunately it's the kind of thing that almost everyone takes for granted and seldom bothers to define explicitly; we might just have to use people's practical usage of the term as implicit evidence for a definition. –jacobolus (t) 00:29, 8 September 2024 (UTC)

I gave it a try, I hope it suffices. fgnievinski (talk) 05:25, 8 September 2024 (UTC)

The way I read the text in that link, "(relative) direction" means the same as what this article is calling "direction", but I find the text in this source to be pretty confused/confusing ("If the points are not fixed in space, it is a displacement vector from the initial point to the final point. [...] A (displacement) vector used to denote the direction from the initial point to the final point is usually called a direction vector. [...] If a vector's initial point is fixed at the origin O, then it is the position vector of the final point relative to the origin O."), and personally I don't think it seems like a very reliable source.

I would instead say something like: "A vector, which can be pictured as an arrow (directed straight line segment) from one point at its tail to another point at its tip, can be used to represent a translation from the first point to the second. In Euclidean space, translations move all points uniformly: each point moves the same distance along a line parallel to the line of motion of any other point, and the distances between points are preserved; therefore, an arrow between an initial and final point of a translation can be moved to begin at any other point in the space and describe the same translation. This type of vector is often called a displacement vector. Sometimes, especially in physics, a vector is used to represent the position of an arbitrary point relative to a designated origin; this type of vector is called a position vector, and can be pictured as an arrow with its tail at the origin and its tip at the given point." –jacobolus (t) 06:38, 8 September 2024 (UTC)

A "bound vector" is defined as a directed line segment, not just depicted as such. The direction of a bound vector is another bound vector, of unit length. We should avoid shoehorning the mechanics concept into Euclidean vectors instead of directed line segments.fgnievinski (talk) 00:38, 9 September 2024 (UTC)

The direction of a bound vector is another bound vector, of unit length. – I am not finding any reflection of this definition in skimming google scholar links. Is this a definition used by someone else, or just your own personal conception? Personally I would consider "a bound vector of unit length" meaning a directed line segment to be nonsensical in most physics contexts, where a directed line segment always has a length measured in concrete physical units such as meters or inches, whereas a "unit vector" is always a dimensionless quantity whose length is an abstract number which does not represent a physical quantity. –jacobolus (t) 00:55, 9 September 2024 (UTC)

Here's an example where "bound vector" is defined as an ordered pair of points, and "free vector" is defined as the equivalence class of bound vectors that can be translated into each-other: Henderson. Here's a different source which doesn't define (per se) the term bound vector but says "An ordered pair of points determines a bound vector." Hervé. I don't think it's at all universal to claim that a vector is a line segment. –jacobolus (t) 01:33, 9 September 2024 (UTC)

I've rewritten the contentious passage with a milder version which just acknowledges there's a related concept (a unit directed line segment), as I suspect it's a common confusion. fgnievinski (talk) 04:17, 9 September 2024 (UTC)

Sorry, I'm not trying to rake you over coals here. I just think we should try to be careful not to make authoritative claims that aren't pretty solidly representative of common usage / the best sources we can find. I'll keep looking around to see if I can find some good survey papers discussing variations in definitions and concepts. I still don't like our text here; I may rewrite it again, if I can find better sources. –jacobolus (t) 05:06, 9 September 2024 (UTC)