Parallel Vectors in R^2: Understanding Magnitude and Direction

[fmadero]

Homework Statement

I have a problem with this statement in a Calculus book:
A scalar multiple $$s\vec{v}$$ of $$\vec{v}$$ is parallel to $$\vec{v}$$ with magnitude $$|s|\ ||\vec{v}||$$ and points in the same direction as $$\vec{v}$$ if $$s>0$$, and in the opposite direction if $$s<0$$

What bothers me about this statement is it never talks about the origin if they have the same origin then how can they be parallel, there will be multiple intersections. If we take $$\vec{v}=\left<2,2\right>$$ and $$\vec{t}=\left<6,6\right>$$ and $$s=\frac{1}{3}$$ then using the statement above $$\vec{v}$$ and $$\vec{t}$$ are parallel if and only if $$\vec{v}=s\vec{t}$$, which they are but if you graph it in 2 dimensions you can easily see if both vectors origin are the same say 0,0 then they are not parallel.

Are we suppose to assume the origin is never the same, unless explicitly told?

 

[tiny-tim]

What bothers me about this statement is it never talks about the origin if they have the same origin then how can they be parallel, there will be multiple intersections.

Hi fmadero!

We can slide vectors anywhere … for example, to make parallelograms to add two vectors.

Since we can slide them, the most we can say about a (pure) vector is its direction and its magnitude, not its origin.

If we take $$\vec{v}=\left<2,2\right>$$ and $$\vec{t}=\left<6,6\right>$$ and $$s=\frac{1}{3}$$ then using the statement above $$\vec{v}$$ and $$\vec{t}$$ are parallel if and only if $$\vec{v}=s\vec{t}$$, which they are but if you graph it in 2 dimensions you can easily see if both vectors origin are the same say 0,0 then they are not parallel.

Sorry … I'm not following that.

if you graph (2,2) and (6,6) in 2 dimensions, they are parallel (in fact, in the same straight line), aren't they?

 

[fmadero]

Hi fmadero!

We can slide vectors anywhere … for example, to make parallelograms to add two vectors.

Since we can slide them, the most we can say about a (pure) vector is its direction and its magnitude, not its origin.


Sorry … I'm not following that.

if you graph (2,2) and (6,6) in 2 dimensions, they are parallel (in fact, in the same straight line), aren't they?

Yes! one is just longer than the other and since one is longer than the other they share common points thus they are not parallel by the definition of parallel lines right?

 

[tiny-tim]

"parallel"

Yes! one is just longer than the other and since one is longer than the other they share common points thus they are not parallel by the definition of parallel lines right?

Nooo … identical, or overlapping, lines are still parallel.

Oh I see what's bothering you … in Euclidean geometry, parallel lines don't meet, by definition, and so they can't share any points.

Forget that … this is vector geometry, and (for practical reasons ) the definition of parallel is different! ​

 

[fmadero]

Nooo … identical, or overlapping, lines are still parallel.

Oh I see what's bothering you … in Euclidean geometry, parallel lines don't meet, by definition, and so they can't share any points.

Yes based Euclidean geometry, http://en.wikipedia.org/wiki/Parallel_lines#Euclidean_parallelism"

Forget that … this is vector geometry, and (for practical reasons ) the definition of parallel is different! ​

Where did you read this from? You recommend any sites or books?

 

[tiny-tim]

Nooo … identical, or overlapping, lines are still parallel.
Oh I see what's bothering you … in Euclidean geometry, parallel lines don't meet, by definition, and so they can't share any points.

Yes based Euclidean geometry, http://en.wikipedia.org/wiki/Paralle...an_parallelism

Forget that … this is vector geometry, and (for practical reasons ) the definition of parallel is different!

oi! don't misquote people! …

I did not quote that wiki reference.

Where did you read this from? You recommend any sites or books?

erm … it's obvious from the definition of a vector …

but see, for example, http://mathworld.wolfram.com/ParallelVectors.html

 

[fmadero]

oi! don't misquote people! …

I did not quote that wiki reference.


erm … it's obvious from the definition of a vector …

but see, for example, http://mathworld.wolfram.com/ParallelVectors.html

Sorry I was trying to interleave my responses, I don't use forums too often.

thanks
frank