Calculating Midpoint of Vector AB | Precalc Homework Solution

[Mathematicsss]

Mod note: Moved to Precalc section

Homework Statement

Find the midpoint of the vector AB[/B]
A(3,2,5) B(1,3,2)

Homework Equations

Not sure

The Attempt at a Solution

I wrote each point in terms of the unit vectors, i,j,k then I subtracted the two, and divided by a 2.

 

[Orodruin]

I wrote each point in terms of the unit vectors, i,j,k then I subtracted the two, and divided by a 2.

In the future, please do not just describe what you did. Show us what you did and what result you got.

Now. Why did you take the difference and divide by two? Why do you think that will give you the midpoint? If you think just about real numbers, say 4 and 10, which number is the midpoint between those and do you get it by taking (10-4)/2?

 

[Mathematicsss]

In the future, please do not just describe what you did. Show us what you did and what result you got.

Now. Why did you take the difference and divide by two? Why do you think that will give you the midpoint? If you think just about real numbers, say 4 and 10, which number is the midpoint between those and do you get it by taking (10-4)/2?

Because vector AB is point B relative to A, and so I subtracted the two to find the new point, and then wrote the new point in the form of i,j and k unit vectors. I then divided in order to the midpoint of the the new vector. However, I am not sure why it is that it doesn't work with the numbers you wrote (4 and 10), can you please explain?

 

[Orodruin]

Because vector AB is point B relative to A, and so I subtracted the two to find the new point, and then wrote the new point in the form of i,j and k unit vectors.

This gives you the difference vector between A and B, not the midpoint. Consider the case when A and B are the same point, you would get zero.

What would you do to mathematically find the midpoint between 4 and 10?
Edit: It is obviously 7, but how do you reach this conclusion?

 

[Mathematicsss]

This gives you the difference vector between A and B, not the midpoint. Consider the case when A and B are the same point, you would get zero.

What would you do to mathematically find the midpoint between 4 and 10?
Edit: It is obviously 7, but how do you reach this conclusion?

To find the midpoint between 4 and 10, I add 4+10 and divide by 2.

 

[Orodruin]

To find the midpoint between 4 and 10, I add 4+10 and divide by 2.

So why are you not doing the same for your problem?

 

[Mathematicsss]

So why are you not doing the same for your problem?

Oh, I see.

 

[Mathematicsss]

So why are you not doing the same for your problem?

I have another question, to write A(3,2,5) as a position vector, it would be 3i+2j+5k correct?

 

[Orodruin]

Assuming that the (3,2,5) are the Cartesian coordinates, yes.

 

[Mathematicsss]

Assuming that the (3,2,5) are the Cartesian coordinates, yes.

What if I was asked just to find the position vector AB?

 

[Orodruin]

AB is not a position vector, it is a difference vector.

 

[Mathematicsss]

AB is not a position vector, it is a difference vector.

Yes, I know.. I meant that what if we were supposed to find a position vector, AB, is that possible? and what if I wrote AB (difference vector) in the form of I,j, and k? wouldn't it be a position vector? Assuming A and B are cartesian coordinates?

 

[Orodruin]

wouldn't it be a position vector?

No. It would only be a vector. The only thing that is a position vector is the vector from the origin to a given point. AB is not such a vector.

 

[Mathematicsss]

No. It would only be a vector. The only thing that is a position vector is the vector from the origin to a given point. AB is not such a vector.

Alright, do you suggest that in order for myself to solve vector problems that are in R^3, it would be helpful to use an analogy in R^2?

 

[Mathematicsss]

No. It would only be a vector. The only thing that is a position vector is the vector from the origin to a given point. AB is not such a vector.

last two questions, 1)if I added the two vectors are wrote them in I,j,and k, will I get a position vector?
2) Will there ever be a case where I will not be able to find the midpoint of AB?

Thank you very much, you have been very helpful. I apologise for taking much of your time. Have an amazing day!

 

[FactChecker]

We need to be careful of the wording of the question. As stated, I think that the vector AB is (-1, 1/2, -3/2). That is the vector originating at (0,0,0). The vector from A to B is exactly the difference as stated in the OP and the midpoint of that vector is obtained as stated in the OP.

If the problem had asked for the midpoint of the line from A to B, I think that would be different. That would be (A+B)/2. Or if the question had asked for the vector from the origin to the midpoint between A and B, that would also be the vector from the origin to the point (A+B)/2 = A + (AB/2)

PS. I'm using bold to indicate vectors because I have never been able to consistently get the vector symbol to work.

 

[Orodruin]

I don't think the question in the OP is very well worded. I suspect it might not be stated exactly as given.

In my mind it does not make sense to ask for the "midpoint" of a vector. You only ask for the midpoint between to given points. If I wanted half the distance vector between the points, I would ask for half of AB. I suspect the problem might have come with a figure indicating AB as the vector from A to B. In that case, the midpoint of the vector would indeed be the midpoint between A and B. Without further specification and the problem stated exactly as given, we simply do not know.

 

[FactChecker]

I don't think the question in the OP is very well worded. I suspect it might not be stated exactly as given.

I agree.

In my mind it does not make sense to ask for the "midpoint" of a vector. You only ask for the midpoint between to given points. If I wanted half the distance vector between the points, I would ask for half of AB.

Good point.

I suspect the problem might have come with a figure indicating AB as the vector from A to B. In that case, the midpoint of the vector would indeed be the midpoint between A and B. Without further specification and the problem stated exactly as given, we simply do not know.

I agree. It's unfortunate that the wording is so treacherous.

 

[LCKurtz]

PS. I'm using bold to indicate vectors because I have never been able to consistently get the vector symbol to work.

Try ##\vec A##. It will render as $\vec A$.

 

[FactChecker]

Try ##\vec A##. It will render as $\vec A$.

Thanks. After some experimenting, I see that it was when I tried to use a bold letter as a vector that it didn't work for me:
Not bold: $\vec A$
Bold: ##\vec A##

 

[Orodruin]

When you insert a bold letter you are in actuality insertin BBcode into your LaTeX expression. If you want to make something bold in LaTeX, you must use LaTeX commands. But in general there is no need to make a vector bold and to have a vector arrow.

 

[Mathematicsss]

I don't think the question in the OP is very well worded. I suspect it might not be stated exactly as given.

In my mind it does not make sense to ask for the "midpoint" of a vector. You only ask for the midpoint between to given points. If I wanted half the distance vector between the points, I would ask for half of AB. I suspect the problem might have come with a figure indicating AB as the vector from A to B. In that case, the midpoint of the vector would indeed be the midpoint between A and B. Without further specification and the problem stated exactly as given, we simply do not know.

The question is as follows: Find the position vector of the midpoint AB.

 

[FactChecker]

The question is as follows: Find the position vector of the midpoint AB.

Aha! Yes, that is different and it is well defined. The midpoint of the line from point A to point B is at (A+B)/2 = ( (3+1)/2, (2+3)/2, (5+2)/2 ) = ( 2, 2.5, 3.5 ). And the vector to that point is based at (0,0,0) and denoted by $\vec{( 2, 2.5, 3.5 )}$.

You can also get it by taking your vector of the OP, $\vec{AB}$, and adding it to the position vector of point A.
$\vec A + \vec {AB} = \vec{(3,2,5)} + \vec{(-1,0.5,-1.5)} = \vec{(2,2.5,3.5)}$

 

[FactChecker]

When you insert a bold letter you are in actuality insertin BBcode into your LaTeX expression. If you want to make something bold in LaTeX, you must use LaTeX commands. But in general there is no need to make a vector bold and to have a vector arrow.

Thanks. I also tried \textbf{}, which didn't work. Maybe I did it wrong. Anyway, I agree that it was not needed and I stopped trying.

 

[Orodruin]

If I have to put bold stuff in equations, I usually use \boldsymbol instead. I think it looks much better as it uses an italic math font instead of text font - also it works for Greek letters. Here is the difference compared to using \textbf or {\bf }:
$$
\boldsymbol{a}\cdot\boldsymbol{\mu} = \textbf{a}\cdot \textbf{\mu} = {\bf a}\cdot{\bf \mu}
$$
Either way, I much prefer using \vec as it looks more similar to what you would write by hand and I think it is just confusing for students that we write things one way in typeset text and another on the blackboard.

 

[Ray Vickson]

Thanks. I also tried \textbf{}, which didn't work. Maybe I did it wrong. Anyway, I agree that it was not needed and I stopped trying.

The commands "\bf{x}" or "\mathbf{x}" both produce $\bf{x}$ and $\mathbf{x}$. The commands "\mathbf{x}^2" and "\mathbf{x^2}" produce $\mathbf{x}^2$ and $\mathbf{x^2}$, one of which has a bold superscript and the other not. The command \mathbf{x \times \alpha}" produces $\mathbf{x \times \alpha}$ (with bold $\bf x$ but non-bold $\alpha$ and non-bold $\times$), while "\boldsymbol{x \times \alpha}" produces $\boldsymbol{x \times \alpha}$, where everything is in bold.