Generating vectors from points.

(vectors are bold)

I have an odd issue I suppose.

Given two points, A and B, and the need to generate a vector from them there are two things I can do, A-B=**C **or B-A=**-C**. So I end up finding **C** when the text/instructor wants **-C**. I have no idea what order to take the points A and B in and I am unaware of a standard. Should I order them by which is the closest to the origin or something?

(If you're confused see Vector Algebra: I think this should link you directly to a section on what method I'm talking about to generate vectors.)

Re: Generating vectors from points.

Quote:

Originally Posted by

**bkbowser** (vectors are bold)Given two points, A and B, and the need to generate a vector from them there are two things I can do, A-B=**C **or B-A=**-C**. So I end up finding **C** when the text/instructor wants **-C**. I have no idea what order to take the points A and B in and I am unaware of a standard. Should I order them by which is the closest to the origin or something?

That is truly a confused statement.

Do you understand that a vector is a quasi mathematical concept in that it is really not defined as most mathematical concepts. A vector is an object that has length and direction. Given points $\displaystyle A~\&~B$ the vector $\displaystyle \overrightarrow {AB} $ represents the direction "moving from A to B" as well as the distance from A to B, $\displaystyle \|\overrightarrow {AB}\| $.

Where as $\displaystyle \overrightarrow {BA} $ represents the direction "moving from B to A" and $\displaystyle \|\overrightarrow {AB}\|=\|\overrightarrow {BA}\| $. They have the same length but opposite direction.

Therefore $\displaystyle \overrightarrow {BA}=-\overrightarrow {AB}$

So it has nothing to do with the point's relative to the origin.

If $\displaystyle P: (a,b,c)~\&~Q: (e,f,g)$ then $\displaystyle \overrightarrow {PQ}=<e-a,f-b,g-c>$ and $\displaystyle \overrightarrow {QP}=<a-e,b-f,c-g>$.

Re: Generating vectors from points.

Ya I know all that stuff already. I was just hoping that there was some sort of broadly accepted standard for how to set points up so that people don't have to constantly deal with negative one scalar multiples of otherwise identical vectors.

Re: Generating vectors from points.

Quote:

Originally Posted by

**bkbowser** Ya I know all that stuff already. I was just hoping that there was some sort of broadly accepted standard for how to set points up so that people don't have to constantly deal with negative one scalar multiples of otherwise identical vectors.

That statement makes absolutely no sense what so ever.

If you truly understand "all that stuff", there would no need to say "so that people don't have to constantly deal with negative one scalar multiples of otherwise identical vectors."

If vectors are negatives of each other then they certainly not identical.

The whole point is the *opposite directions*.

Re: Generating vectors from points.

I think that your assuming that I'm given an A and a B in the context of a problem that reads something like "given the points A and B find some vector C." Or, there is something about movement between positions in the problem to inform my decision. This is not what's happening.

I'm usually deducing two distinct points P(a,b,c) and P(x,y,z). Since the problem doesn't tell me which way the vector is supposed to go I have the freedom to say arbitrarily that P(a,b,c)=A and P(x,y,z)=B OR that P(a,b,c)=B and P(x,y,z)=A. Does this make sense to you?

Re: Generating vectors from points.

Quote:

Originally Posted by

**bkbowser** I'm usually deducing two distinct points P(a,b,c) and P(x,y,z). Since the problem doesn't tell me which way the vector is supposed to go I have the freedom to say arbitrarily that P(a,b,c)=A and P(x,y,z)=B OR that P(a,b,c)=B and P(x,y,z)=A. Does this make sense to you?

No it makes absolutely no sense to anyone.

You simply do not understand points, much less vectors.

If $\displaystyle (a,b,c)$ is a point and $\displaystyle (x,y,z)$ is a point then of course you are free to name the two any thing you wish.

But given $\displaystyle P: (a,b,c)~\&~Q: (x,y,z)$ are two points then they come named. Therefore, you may not rename them. The example you gave does not occur. Once two points are labeled, now we can address vectors.

Now it is true that often we say $\displaystyle \vec{a}$ is a vector.

We don't have two point specified. But the ideas of *length and direction* are still there.

Consider two pairs of points $\displaystyle P: (2,-3,5)~\&~Q: (5,-6,1)$ and the pair $\displaystyle A: (-1,3,2)~\&~B: (2,0,-2)$.

There are **four different points**.

BUT $\displaystyle \overrightarrow {PQ} = \overrightarrow {AB} $, **one vector**.

A vector is an equivalence class of objects that have the same length and direction.

Re: Generating vectors from points.

Quote:

Originally Posted by

**Plato** No it makes absolutely no sense to anyone.

You simply do not understand points, much less vectors.

If $\displaystyle (a,b,c)$ is a point and $\displaystyle (x,y,z)$ is a point then of course you are free to name the two any thing you wish.

But given $\displaystyle P: (a,b,c)~\&~Q: (x,y,z)$ are two points then they come named. Therefore, you may not rename them. The example you gave does not occur. Once two points are labeled, now we can address vectors.

Now it is true that often we say $\displaystyle \vec{a}$ is a vector.

We don't have two point specified. But the ideas of *length and direction* are still there.

Consider two pairs of points $\displaystyle P: (2,-3,5)~\&~Q: (5,-6,1)$ and the pair $\displaystyle A: (-1,3,2)~\&~B: (2,0,-2)$.

There are **four different points**.

BUT $\displaystyle \overrightarrow {PQ} = \overrightarrow {AB} $, **one vector**.

A vector is an equivalence class of objects that have the same length and direction.

**so that people don't have to constantly deal with negative one scalar multiples of otherwise identical vectors.**

Most of your post leads me to believe that you think this bold sentence means A=-A. It doesn't.

What I'm saying is that there are sometimes no contextual reasons for me to use the vector A. Just as there are sometimes no contextual reason for me to use the vector -A. So I get to pick one or the other.

So I am **not** in this situation;

Quote:

But given $\displaystyle P: (a,b,c)~\&~Q: (x,y,z)$ are two points then they come named. Therefore, you may not rename them. The example you gave does not occur. Once two points are labeled, now we can address vectors.

**But rather this situation;**

Quote:

If $\displaystyle (a,b,c)$ is a point and $\displaystyle (x,y,z)$ is a point then of course you are free to name the two any thing you wish.

So there's no reason to talk about the first situation.

My problem is that the Textbook/Instructor are in the same situation I am and are free to choose A instead of -A. I was hoping there was some standard to prevent this confusion just to make my solutions easier to understand.