Whats the meaning of Parallel in Vectors

[Arman777]

Lets suppose we have a two vectors where $\vec u=c\vec r$ where c is just a reel constant number.Can we say $\vec u$ and $\vec r$ is parallel.

How can we define ""parallel" vectors ? Like in most general way.

I know that when c is positive real number they are definitely parallel.But when c is negative still can we call them "parallel".I am thinking yes we can but I need some solid proof

Thanks

 

[Orodruin]

Proof? It is a matter of definition ...

 

[Arman777]

Proof? It is a matter of definition ...

Solid proof of definition.I don't even know that make sense or not but I mean like a definition comes from some general acceptable textbook or from some article etc.Not just an idea but I can show some referance.

 

[fresh_42]

1. parallel vectors = parallel lines, defined by these vectors
2. parallel lines = constant distance at each point

So now, you have to define the distance between two lines. It is here, were the definition plays a role, because there is no universally valid concept of distance. That's why we speak of metrics. Tell me your metric and I will tell you parallelism.

 

[Arman777]

ell me your metric and I will tell you parallelism.

Metric ? I don't know ...

 

[Arman777]

Think like this for simple case. $\vec u=(1,0,0)$ and $\vec r=(-1,0,0)$ Are these vectors parallel ?

 

[fresh_42]

In Euclidean space, yes. However, you could either define "parallel" by "constant distance", which I prefer to use, or by "constant distance greater than zero", in which case they wouldn't be parallel. After my fancy, this is an ugly condition, but Euclid probably used this distinction.

 

[Arman777]

In Euclidean space, yes. However, you could either define "parallel" by "constant distance", which I prefer to use, or by "constant distance greater than zero", in which case they wouldn't be parallel. After my fancy, this is an ugly condition, but Euclid probably used this distinction.

Makea sense..I don't know that much of geometry/algebra sadly.But I understand.
Thanks :)