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It seems to me that perpendicular, orthogonal and normal are all equivalent in two and three dimensions. I'm curious as to which situations you would want to use one term over the other in two and three dimensions.

Also... what about higher dimensions? It seems like perpendicular and normal would not have a nice meaning whereas orthogonal would as it is defined in terms of the dot product.

Can someone give me a detailed breakdown as to the differences in their meanings, their uses and the situations for which each should be used?

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migrated from mathematica.stackexchange.com Aug 25 '17 at 22:15

This question came from our site for users of Wolfram Mathematica.

  • $\begingroup$ Is this a Mathematica question?? $\endgroup$ – Carl Woll Aug 25 '17 at 22:14
  • $\begingroup$ No. This is a mathematics question. $\endgroup$ – Michael McCain Aug 25 '17 at 22:15
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    $\begingroup$ Orthogonal is likely the more general term. For example I can define orthogonality for functions and then state that various sin () and cos () functions are orthogonal. An orthogonal basis can be used to decompose something into independent components. For example, the Fourier transform decomposes a time domain function into weights of sines and cosines. A triple in 3D space is a decomposition of a vector in 3D space along 3 orthogonal basis vectors. $\endgroup$ – Andy Walls Aug 25 '17 at 22:27
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    $\begingroup$ @AndyWalls I tend to agree that “orthogonal” includes “perpendicular.” The zero vector is orthogonal to everything, but I don’t think that I’d call a zero-length vector “perpendicular” to anything. $\endgroup$ – amd Aug 25 '17 at 23:57
  • $\begingroup$ Don't forget when lines have unit spread ($\sin^2 \theta = 1$) also. $\endgroup$ – ja72 Aug 26 '17 at 1:07
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Excellent Question! The best way to under stand the three (3) terms is in the context of the History of Mathematics. Also, the notion of Logical Equivalence is also useful in our understanding. Think of the various ways in which the Parallel Lines Axiom can be replaced by other Statements which turn out to be Logically Equivalent. So,

(1) Perpendicular is usually associated with "dropping a perpendicular from a point," and it only presupposes 2 Dimensions in the form of a Single Plane. And no Angularity (Angles) are assumed. For example, Given a Line (Horizontal) with 2 Congruent Circles intersecting at 2 Points) on it (the Centers lie on the Line), the line "Dropped" from the Top to the Bottom, forms a Perpendicular. Notice that nothing is said about Angles.

(2) Orthogonal Line is defined as a line with a Right Angle. And since all Right Angles are Equal (one of Euclid's Axioms) every Orthogonal Line implies 4 Rays from the point of intersection.

Notice that one can prove (probably, depending on who you are and which axioms you choose) that every Orthogonal is a Perpendicular, and every Perpendicular is an Orthogonal.

(3) The Normal is a Perpendicular to a Plane Tangent to a Surface. So at least three (3) Dimensions. Generalizing, every Pair of Dimensions produced a distinct "new' Plane, an for each a Normal may or can be Defined which lies in the "next" dimension of the given 2 - the restriction is that the Normal only has One Point common to the Plane it intersects, so it lies in the "other" Dimension than the 2 which it is Perpendicular or Orthogonal to. Any Ray from the Point to which the Normal is Defined forms a Plane, and in that Plane the Normal forms a Perpendicular which is also an Orthogonal. Adding Dimensions is merely adding new Panes wherein a new Perpendicular(s), Orthogonal(s), and Normal(s0 are introduced.

Hope this answers your excellent question.

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In two or three dimensions, I agree, perpendicular is more natural than orthogonal.

In higher dimensions, or if the dimension is represented by an unknown, both are correct, but I think orthogonal is preferable.

Here's an excerpt from Wikipedia (https://en.wikipedia.org/wiki/Orthogonality):

In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms.

Normal can be used in any dimension, but it usually means perpendicular to a curve or surface (of some dimension).

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Perpendicular lines may or may not touch each other. Orthogonal lines are perpendicular and touch each other at junction.

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  • $\begingroup$ hmmm... I don't think orthgonal lines have such condition. $\endgroup$ – Siong Thye Goh Sep 15 '18 at 12:31

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