# What does 'orthogonal' mean???

by urbantrained
Tags: orthogonal
 P: n/a ????
 Sci Advisor P: 875 The simplest definition is "forming a right angle". When you get to multidimensional space, vectors are orthoganal if they have finite magnitude and no projection on each other. It is also used figuratively. Ideas are orthoganal if they take different but non-competing paths to solving the same problem. Njorl
 Sci Advisor HW Helper P: 1,772 THe word "perpendicular" has replaced "orthogonal" in everyday use. Perpendicular is supposed to mean specifically "vertical," but it has come to mean "at a right angle to." "Normal" is another word that often replaces orthogonal in physics. I think that "orthogonal" is still the preferred word used by mathematicians? Calculus was the only place I remember using it.
P: 875

## What does 'orthogonal' mean???

I always hated that use of "normal". Later, you get into probabiity functions where normal has a different meaning - that the sum of all probabilities adds up to one. Such a function is normalized. Then there are ortho-normal bases, a sets of mutually orthoganal unit vectors that completely define a space. That other definition of normal makes them a little more confusing.

Just one of my pet peeves. That, and using gothic letters for statistical mechanics. [!:)]

Njorl
 P: 454
HW Helper
P: 1,772
 Originally posted by Njorl Such a function is normalized. Then there are ortho-normal bases, a sets of mutually orthoganal unit vectors that completely define a space. Njorl
OWWW!

You just revived a twenty year old pain at the top of my head!
P: 875
 Originally posted by Chi Meson OWWW! You just revived a twenty year old pain at the top of my head!
Heh. Twenty one years ago this week I took my Linear algebra final.

Njorl
Mentor
P: 7,290
The formal definition of Orthogonal is

$$A \cdot B = 0 \mbox{and} |A| > 0, |B|>0$$

To me normal is 3d perpendicular, a line is NORMAL to a plane but it is perpendicular to a second line.

Example, if I balance a pencil on its tip on a desk top the pencil is NORMAL to the desk. If I lay 2 pencils on the desk top and arrange them so the form a right angle they are perpendicular. Perpendicular => 2d Normal =>Nd (N>=3)

 Then there are ortho-normal bases, a sets of mutually orthogonal unit vectors that completely define a space.
This is just an application of Normal in the sense of the 3d Normal, it simply extends the idea to even higher dimensions.

I on the other hand have always liked the term "Normal". Cus when I drill a hole at some weird angle, I can say "Well I have never been Normal" [:)]

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