# Prove RS is parallel to KL using vector method

• Richie Smash
In summary: Ummm NO sir, that would mean then that, KL, or in other words KB is, : -K/2 +M/2-K/4So in other words that would be -3/4K +M/2 is...That is correct.
Richie Smash

## Homework Statement

Hi I have attached an image which shows OK and OM which are position vectors such that OK=k and OM =m
R is the mid point of OK and S is a point on OM such that (vec) OS = 1/3 OM

L is the midpoint of RM, using a vector method, prove RS is parallel to KL.

## The Attempt at a Solution

SO the questions asked me to find these vectors

MK=k-m

RM= m-k/2

KS= m/3-K

RS= m/3-k/2

So now tht I've found these vectors,

I would think that I need to prove that KL and RS are multiples of one another to show they are parallel.

First I found the coordinates of L which is the midpoint of RM

L= (m-k/2)/2= m/2-k/4

So now that I know L... I find KL which would be : -k+(m/2-k)=m/2-2k

SO I know KL, and I know RS which is : m/3-k/2

But they don't seem to be multiples of each other... can anyone help me?

#### Attachments

• vectors 2.png
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You are treating the vectors as scalars, so your length calculations are incorrect. You need to consider the angles between the vectors and use the law of sines and law of cosines to get distances MK, RM, etc. Or better yet, use trigonometry to get the x- and y-coordinates of each point. Then you can use the Pythagorean theorem to get the distances.

Hey tnich, I don't doubt that what youre saying isn't correct, not at all, but I don't think that this level I'm doing requires that, as they would usually give angles it it was required, I'm pretty sure there has to be another way.

Richie Smash said:
Hey tnich, I don't doubt that what youre saying isn't correct, not at all, but I don't think that this level I'm doing requires that, as they would usually give angles it it was required, I'm pretty sure there has to be another way.
Good luck with that.

This step doesn't look right:
First I found the coordinates of L which is the midpoint of RM

L= (m-k/2)/2= m/2-k
When you distribute in the 1/2, (-k/2)*(1/2) should be -k/4, right?

Oh yes it should be -k/4

You are on the right track but your notation is awful.

Richie Smash said:

## Homework Statement

Hi I have attached an image which shows OK and OM which are position vectors such that OK=k and OM =m
R is the mid point of OK and S is a point on OM such that (vec) OS = 1/3 OM

L is the midpoint of RM, using a vector method, prove RS is parallel to KL.

## The Attempt at a Solution

SO the questions asked me to find these vectors

MK=k-m

RM= m-k/2

KS= m/3-K

RS= m/3-k/2

So now tht I've found these vectors,

I would think that I need to prove that KL and RS are multiples of one another to show they are parallel.

First I found the coordinates of L which is the midpoint of RM

L= (m-k/2)/2= m/2-k
Check your arithmetic right there. Should get m/2 - k/4.

What... do I do now?

In this picture I have labeled points with lower case and vectors with upper case. So:
##\vec K = ok,~ \vec R = or = \frac 1 2 \vec K,~\vec M = om,~ \vec S = os = \frac 1 3 \vec M## so ##rs=\vec S -\vec R
=\frac 1 3 \vec M - \frac 1 2 \vec K##, which you have correct above.
Also ##\vec B = \frac 1 2 rm = \frac 1 2(\vec M - \vec R) ## to get you going. Express that in terms of ##\vec M##
and ##\vec K##, then get ##kl## in terms of them too. Show us how that goes.

#### Attachments

• vectors.jpg
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Richie Smash
I believe that B would be M/2 -K/2

But hmm since RS is M/3-K/2

KL must be M/4+2K/8

Richie Smash said:
I believe that B would be M/2 -K/2

Show your work. How do you get that from ##\vec B=\frac 1 2(\vec M - \vec R)##?

This is quite confusing, but to express B=1/2(M-R)

R is the same as K/2

So in that sense B would be M/2-K/2 yes?

Richie Smash said:
This is quite confusing, but to express B=1/2(M-R)

R is the same as K/2

So in that sense B would be M/2-K/2 yes?
No. Same kind of algebra mistake you had before.

What mistake might that be? having the new diagram clears up the position of the lines for me but I still find it a bit difficult

It's the same mistake you made earlier. Look back at the earlier posts.

Oh yes I see, B would be M/2-K/4

So is that all you have to tell me?

Ummm NO sir, that would mean then that, KL, or in other words KB is, : -K/2 +M/2-K/4

So in other words that would be -3/4K +M/2 is KL

Richie Smash said:
Ummm NO sir, that would mean then that, KL, or in other words KB is, : -K/2 +M/2-K/4

So in other words that would be -3/4K +M/2 is KL
So ...?

Well at this point I fail to see how RS or Kl could be a multiple of one another because if RS = M/3-K/2 then I just don't see the correlation unless I use common fractions then RS would be 4/12M-6/12K

and KL would be 6/12 M -9/12K then I suppose that KL is 1.5RS?

This does seem correct...

Yes, it does seem correct. I have a few parting comments though. First, there is no need for decimals in this problem. You should be able to show directly that ##kl = \frac 3 2 rs##. Also note that you are confusing points with vectors when you write KL when you should write ##kl##. Finally, you are also abusing mathematics notation when you write something like 4/12M when you mean (4/12)M. Hopefully with more practice you will get better at it.

Richie Smash

## 1. How do you use the vector method to prove parallel lines?

The vector method uses the properties of vectors to show that two lines are parallel. This method involves finding the direction vectors of each line and showing that they are either equal or proportional to each other.

## 2. What are the steps for proving parallel lines using the vector method?

Step 1: Identify the direction vectors of each line by choosing two points on each line and finding the vectors between them.

Step 2: Show that the direction vectors are equal or proportional to each other. If they are equal, then the lines are parallel. If they are proportional, then the lines are parallel or lie on the same plane.

Step 3: If the direction vectors are equal, use the fact that parallel lines have equal slopes to prove that the lines are parallel.

## 3. What is the difference between using the vector method and traditional methods to prove parallel lines?

The traditional method uses the properties of parallel lines, such as equal opposite angles or congruent alternate interior angles, to prove parallel lines. The vector method, on the other hand, uses the properties of vectors to show that two lines have the same direction and therefore are parallel.

## 4. Can the vector method be used to prove parallel lines in three-dimensional space?

Yes, the vector method can be used to prove parallel lines in any dimension. It is not limited to only two-dimensional space.

## 5. Are there any limitations to using the vector method to prove parallel lines?

One limitation is that the vector method can only be used to prove parallel lines that lie on the same plane. It cannot be used to prove parallel lines that are skew, meaning they do not intersect and are not on the same plane.

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