When it comes to using relational operators on scalar values, there's usually no problem. But what about vectors?

 

Are there any generally accepted rules or consensus when it comes to relational comparisons (ie. =, <>, >, <, >=, <=) of vectors?

 

For instance, consider v1 < v2 for the two vectors v1 = (10, 8) and v2 = (5, 10)...

 

Does one then compare the x and y values pairwise and return a boolean for each value, in this case (false, true)?

 

Or does one check the magnitude of the vectors against each other and simply return (false)?

 

Or something entirely different...?

 

Cheers,

 

Michael

I don't think I've ever seen relational operators used on vectors, and don't exactly know what would be meant by v > w. Maybe, somewhere out there, someone has defined the relationships, but there is no such thing in common use today. Like you said, you can always compare components of each vector or their magnitudes, but nothing general has been defined that I am aware of.

For instance, consider v1 < v2 for the two vectors v1 = (10, 8) and v2 = (5, 10)...

 

That's called a lexical ordering, very useful for comparing the sequence ('S', 'm', 'i', 't', 'h') to the sequence ('J', 'o', 'n', 'e', 's'). It's not very useful for comparing vectors. A lexical ordering is a kind of partial ordering. Much more useful is a total ordering such as for the reals. However, there is no way to develop a total ordering for Rⁿ for any n>1 (e.g., the complex numbers).

There are ways, but they're not very nice. These include the dictionary ordering - check the first coordinate, and use it if the two are not equal. If not, go to the second coordinate, etc.

=Uncool-

On a vector space the only thing you can do in general is say that they are equal. This is because we have a well defined zero-vector, 0. The vectors a,b are equal iff

 

a-b = 0

 

This is then understood as the components being equal. (you should then show that it does define an equivalence)

 

Generally there is no product or inner product between vectors (or more correctly for inner product between vectors and dual vectors).

 

As the magnitude of a vector requires a inner product (or a metric) generally we do not have the notion of a magnitude.

cool thx ^^