Are parallel vectors always in the same direction?

[venomnert]

I am just starting to learn about vectors and I have question regarding vectors and their parallel properties:

Assume that we have two vectors A and B, and they are said to be parallel. We know that in order to be parallel the angle between the two vectors either have to be 0 or 180. However, after consulting StackOverFlow, I am told that "Two vectors are parallel if they have the same direction". Thus, eliminating the fact that the angle between the vectors can't be 180.

So my question is, can someone verify the statement, if two vectors are in the same direction then they are parallel and have an angle of 0.

 

[Doc Al]

So my question is, can someone verify the statement, if two vectors are in the same direction then they are parallel and have an angle of 0.

That statement is certainly true. But I believe the most useful definition of "parallel vectors" would allow them to have an angle of 0 or 180. (Some call vectors with an angle of 0 parallel and those with an angle of 180 anti-parallel.)

What's the physics context of your question?

 

[venomnert]

That statement is certainly true. But I believe the most useful definition of "parallel vectors" would allow them to have an angle of 0 or 180. (Some call vectors with an angle of 0 parallel and those with an angle of 180 anti-parallel.)

What's the physics context of your question?

Okay, so for the statement "A unit vector is labeled by a caret; the vector of unit length parallel to A is A(caret)". The unit vector of A is in the same direction as A since it 's parallel. Had they said it was anti-parallel, the angle would be 180 and the vectors would be opposite in direction.

 

[David Lewis]

The unit vector and vector A are collinear.

 

[Kajal Sengupta]

We know that in order to be parallel the angle between the two vectors either have to be 0 or 180.

I don't think that this statement is correct. When angle between two lines ( not vectors) is either 0 or 180 then they are collinear. When we come to vector then we need to understand that vectors are not mere straight lines . They have a direction too. So if angle between two vectors is 180 they are not parallel but oppositely directed and hence called anti-parallel. One need to understand that there is a difference between Maths and Physics. Take the example of two men one traveling towards North and other towards South. If you say that their paths are parallel then how do you distinguish between them?

 

[venomnert]

Take the example of two men one traveling towards North and other towards South. If you say that their paths are parallel then how do you distinguish between them?

In term of Physics, since they are both traveling the opposite direction they are anti-parallel.

 

[Doc Al]

Take the example of two men one traveling towards North and other towards South. If you say that their paths are parallel then how do you distinguish between them?

Well, they have different directions!

In term of Physics, since they are both traveling the opposite direction they are anti-parallel.

That's fine.

But it is perfectly reasonable to call those two men as traveling in parallel paths. (I would not get too hung up on this terminology.)

 

[Khashishi]

Well, they would be traveling along parallel lines, but the vectors are not parallel.

 

[Doc Al]

Well, they would be traveling along parallel lines, but the vectors are not parallel.

Why not? What definition of "parallel" are you using?

 

[Kajal Sengupta]

Why not? What definition of "parallel" are you using?

My take is that when you use the word parallel in context of Maths then two lines having an angle between them as 180 can be called parallel more correctly co-linear. In Physics the nuances of the same word is lightly different and in such a case we call two vectors as anti-parallel. Don't know whether it is acceptable to you.