Difference between Perpendicular, Orthogonal and Normal

Question

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?

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. — Andy Walls, Aug 25 '17 at 22:27
@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. — amd, Aug 25 '17 at 23:57
Don't forget when lines have unit spread ($\sin^2 \theta = 1$) also. — John Alexiou, Aug 26 '17 at 01:07

quasi · Answer 1 · 2017-08-25T22:37:02.583

18

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).

edited Aug 25 '17 at 22:37

answered Aug 25 '17 at 22:25

quasi

58,772

The Wikipedia article for orthogonal matrix suggests that its vectors must also be normalized. This is confusing to me, because I thought this is what the term orthonormal is for. Can somebody help clarify? Or, fix the name on Wikipedia? https://en.wikipedia.org/wiki/Orthogonal_matrix – Aaron Franke Oct 14 '23 at 18:46

score 12 · Accepted Answer · edited Oct 18 '19 at 17:28

Excellent Question! The best way to understand the three (3) terms is in the context of the History of Mathematics. Additionally, 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(s) are introduced.

Hope this answers your excellent question.

So in short: perpendicular = orthogonal in reference to some other line in 2D, whereas a normal is in reference to a surface in 3D. — Angelorf, Aug 06 '21 at 18:15

score 1 · Answer 3 · answered Oct 13 '21 at 05:42

In linear algebra, "perpendicular" is mentioned for non zero vectors whereas "orthogonality" is for zero and non zero vectors. We say a zero vector is orthogonal to some vector, not perpendicular. "Normal" is perpendicular to every vector on the plane

score 0 · Answer 4 · answered Jul 25 '19 at 03:43

0

Two sub-spaces F and G (from Euclidean space E finite dim) are perpendicular if they are orthogonal and sums to the whole space i.e F⊕G=E for example two lines are perpendicular in the plane (2D Space) but they are only orthogonal in the 3D space.

answered Jul 25 '19 at 03:43

Anwar AL_ASSIMI

1

score 0 · Answer 5 · answered Jun 01 '22 at 00:35

0

It is important to note that in computer graphics, orthogonal view implies the absence of perspective transforms.

answered Jun 01 '22 at 00:35

Dr. Jay

1

This does not provide an answer to the question. Once you have sufficient reputation you will be able to comment on any post; instead, provide answers that don't require clarification from the asker. - From Review – Integrand Jun 01 '22 at 01:08
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As it’s currently written, your answer is unclear. Please [edit] to add additional details that will help others understand how this addresses the question asked. You can find more information on how to write good answers in the help center. – Community Jun 01 '22 at 01:34

score 0 · Answer 6 · answered Jun 05 '23 at 10:15

Although I appreciate the accepted answer and it includes some interesting insight, right now I was reading Gilbert Strang's Linear Algebra and Learning from Data, and saw this in section I.5, page 29:

The word orthogonal appears everywhere in linear algebra. It means perpendicular. Its use extends far beyond the angle between two vectors.

So I guess, if we trust Gilbert Strang, that I personally do, they are the same, no difference at all.

Steve White · Answer 7 · 2023-09-03T10:52:56.280

Two planes in 3-space are called perpendicular, if any one line in one plane is at right angles to all lines in the other plane. Those two planes share a a line.

But some lines in each of the perpendicular planes are not at right angles to certain lines in the other plane.

In 4-space, for two 2-dimensional planes through the origin to be orthogonal, means that all nonzero vectors in one plane are at right angles to all nonzero vectors in the other plane. Such planes intersect only at the origin.

In this usage, perpendicularity is just a weaker condition than orthogonality.

score -3 · Answer 8 · edited Sep 15 '18 at 15:43

-3

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

edited Sep 15 '18 at 15:43

amWhy

209,954

answered Sep 15 '18 at 12:07

anil kumar

9

2

hmmm... I don't think orthgonal lines have such condition. – Siong Thye Goh Sep 15 '18 at 12:31
You're thinking of `parallel' – Angelorf Aug 06 '21 at 18:16

Difference between Perpendicular, Orthogonal and Normal

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