Difference between Perpendicular, Orthogonal and Normal

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?

Solution 1:

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).

Solution 2:

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.