The significance of orthogonal relationships

[linuspauling]

what is the meaning of orthogonal relationships in addition to right angles in the xyz coordinate system?

for instance, if a 3 piece rocket separated in space in an orthogonal way...will there be any significance when compared to the 3 piece rocket that does not separate in an orthogonal way?

I'm sure pi is somewhere along the answer to this question...am i right?

 

[HallsofIvy]

I have no idea what you mean by "orthogonal relationships in addition to right angles"! "Orthogonal" means "at right angles". I also don't know what kind of "signifcance" you are looking for in your example. There may be some physical significance but not mathematical.

 

[linuspauling]

I just find it a bit fascinating that values can be interpreted in a geometric way. For Instance, the area vector is orthogonal to two vectors in the xy plane after we adjust these two vectors in the xy plane to be orthogonal by projecting one vector over the other...thus, we again form an orthogonal relationship between two sides of rectangle/square that is produced by the two adjusted vectors in the xy plan and the area.

Isn't there some type of meaning here? two vectors in the xy plane form two sides of a geometric shape, and the Area vector is not a geometric shape...thus, it is in another plane...but in an orthogonal way? Am I making any sense? I guess I'm trying to figure out how the originator of vector calculus was constructing this line of logic.

 

[smallphi]

The usual high school geometry is the so called Euclidiean geometry. It is based on axioms. The type of geometry we live in or equivalently the axioms it is based on are verifiable only by experiment.

Actually the Euclidean geometry is just a very good approximation to the geometry on Earth, General relativity predicts that every material body curves the geometry around it so it is no longer Euclidean but the effect on Earth is too tiny so we can usually get away with Euclidean geometry constructions.

 

[linuspauling]

LOOK...determinants work because there are orthogonal relationships between two vectors in the xyz coordinate and the area vector that the parallelogram or the rectangle the two vectors can form.

what is the importance of a 90 degree angle in geometry? sin, cos, tan all work within a 90 degree construction. when working with degrees...pi eventually comes into the picture. who can connect this choppy line of logic and smooth it out?

 

[smallphi]

The sum of all angles in a triangle is 180 deg only in Euclidean geometry. Many of the high school trig formulas are true only in Euclidean geometry. So what is your question? Why we live in an approximately Euclidean world? I doubt anyone knows that.