What is the difference between orthogonal, complementary, null, perpendicular, and such?
from sopularity_fax@sopuli.xyz to nostupidquestions@lemmy.ca on 01 Sep 07:25
https://sopuli.xyz/post/32981683
from sopularity_fax@sopuli.xyz to nostupidquestions@lemmy.ca on 01 Sep 07:25
https://sopuli.xyz/post/32981683
What is the opposite of orthogonal?
Lot of geometry references and I’m having trouble getting a grasp of the essence of orthogonality, it seems.
Also keep seeing perpendicular but what the heck does that mean in conceptual or real life analogies? Wall and floor but again, what the heck does that mean in practical terms?
I initially thought it was like two points being on a same axis but not the same pair of coordinates or in conceptual terms, two things that are points on the same spectrum/axis like in a range from 1-100 or something but it seems like its on two arbitrary points on like a seperate x/y axis so like what is the point of it or where is the actual relationship?
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Roughly speaking, things are orthogonal if they’re independent of each other. In math: at right angles to each other, or, equivalently, the dot product is equal to zero. Are you encountering this in a linear algebra or vector calc class? (If so and you don’t get the relevance, it can be helpful to think of the “shadow” one vector would cast onto another… I can elaborate if this is of interest.)
In the metaphorical sense, the concerns are separate; a shirt doesn’t also get “more red” if you get it in a larger size… The size and color are orthogonal issues when you’re picking one out.
Orthogonal just means at right angles.
The way this is defined if you’re a mathematician is two directions are orthogonal to each other if you can move as much as you like in one direction without changing the location with respect to the other direction.
So it’s like North and East. You can walk east as much as you like without changing how far north you are. But if you have a direction like South East or North West, when you walk that way you will change how far north you are.
We can make this 3d by adding a new direction that is orthogonal to both North and East. This direction would be up or down. Because you can move up and down without changing how far north or east you are.
Mathematicians measure orthogonality of directions using an inner product. If you represent the directions as vector then the inner product between them is zero if they are orthogonal.
Perpendicular means orthogonal, the opposite is parallel.
Null is more confusing. To define this we need to talk about linear functions. A linear function is something that would e.g. create a map from the 3d world, by throwing away up and down information and shrinking everything down so you have a scaled representation of where things are in terms of North and East.
The null space is all the information that is destroyed. For linear functions this is a linear subspace(a space that goes through zero and is described by a collection of directions) that is orthogonal to the space of all information which is kept.