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Can I find the direction of $2$ $n$-dimensional vectors (only $2$ vectors), clockwise or anticlockwise.

For example, If I have $2$ vectors $a=(1,1,1,1,1)$, $b=(2,3,5,6,7)$, function angle(a, b) returns $30°$, and angle $ (b, a) $ will return $-30$°.

I know that the direction can be defined in 2 dimensional space by $x_1y_2 - x_2y_1$.

Can I map the $2$ n-dimensional vectors to $2$ $2$-dimensional vectors(keeping the angle between $2$ vectors not changed or the position not changed between each other) and then check the direction by $2$-dimensional ways? OR any way to find the plane defined by the $2$ $n$-dimensional vectors(From my view, even if they are defined in n-dimensional space, but only $2$ vectors, if the $2$ vectors represent for $2$ intersected lines in n-dimensional space, can I find a $2$-dimensional space(plane) containing the $2$ lines so that I can check the angle and direction in $2$-dimensional ways, like $x_1y_2 - x_2y_1$)?

From my understanding, we can know the direction, clockwise or not in $2$-dimensional space is that we can got the vectors related position by some ways, like using sin/cos and also the formula mentioned, these ways check all the conditions in four quadrant.

But in n-dimensional space, it's hard to compare. Can I map the $2$ $n$-dimension vectors to $2$-dimensional vectors without information losing since in most cases, dimensional reduction will lose information(in this case, if $2$ $n$-dimensional vectors stand for $2$ lines, they can completely present by $2$ $2$-dimensional lines)? Or can I compare the raw n-dimensional vectors directly to check the related position between each other, such as if $a[0] == b[0]$ (or maybe the projection is the same), then we can get the direction information, if not, compare another dimension between the vectors

Above discussion may be not right. Please correct me or any other right ways to do this.

Appreciate for your help

user
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qingdaojunzuo
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  • What is the meaning of the sign in high dimension ($n>2$) case? – user Jun 10 '20 at 06:57
  • @user, thanks for the response. I may not understand the question well. From my understanding, I have 2 n-dimensional vectors, even if they are defined by n-dimensional, but only 2 vectors, if these 2 vectors represent 2 intersected lines in n-dimensional space, can I find a plane contains these 2 vectors and then check the direction by 2-dimensional ways (such as $x_1y_2 - x_2y_1$)? – qingdaojunzuo Jun 10 '20 at 07:40
  • But this is ambiguous. Observe that already in three dimensions you can look at a pair of vectors from opposite sides wrt. the common plane of the vectors with the opposite result for the "product" (clockwise vs. counterclockwise). – user Jun 10 '20 at 08:33

1 Answers1

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In short, you can perform the computation for any number of dimensions greater than 2, and so your answer is essentially answered by this thread. To do the computation, you should be able to compute the norm of a vector and have some familiarity with vector products (dot products and cross products).

What it ultimately boils down to is the right-hand rule in most cases for determining the "direction" or "sign" of the angle once you've computed its magnitude. That said, the sign of the angle is determined arbitrarily to fit an application.

Also, the generalisation of the vector products to higher dimensions has been previously addressed in these threads:

  1. "Geometric definition of the dot product in n-dimensional vector spaces,"

  2. "Cross product in higher dimensions."

In terms of thinking about the plane on which the vectors lie, a plane will always have two dimensions, and so as long as your space has at least two dimensions, you may embed a plane in it. That is, even if you have a 500-dimensional space, a plane will only span a two-dimensional subspace out of the 500 available. If this concept is a bit mind-boggling, I would strongly recommend researching vector spaces, linear combinations, and linear span. Here are some good introductory resources I've found for this topic:

  1. YouTube Playlist: Essence of Linear Algebra by Grant Sanderson (3Blue1Brown)
  2. YouTube Playlist: MIT 18.06 Linear Algebra, Spring 2005 by Gilbert Strang (MIT OpenCourseWare)

I hope you found this helpful or clarifying in some way.

P.S. the tag "dimensional-analysis" is probably not appropriate for this post, as that is the analysis of base units for physical quantities. (See Wikiepdia - Dimensional Analysis.)

Hal
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  • thanks for the detailed reply. I will check these topics in advance. I see that the tag "dimensional-analysis" has already been removed :). Thanks again – qingdaojunzuo Jun 12 '20 at 01:08