Coordinate Geometry: Point Division in Ratio 2 : 5

[roger]

Hi

i'm stuck on this question .
please could someone guide me in the right direction ?


A = (1, - 2,4) and B = (8,12, - 3).

Find the co-ordinates of the point which divides AB in the ratio 2 : 5.



thanx


roger

 

[dextercioby]

Do you have any idea to tackle this problem??Are u familiar with the theorem of similar triangles??

Daniel.

 

[roger]

Do you have any idea to tackle this problem??Are u familiar with the theorem of similar triangles??

Daniel.

Firstly, i was confused as to what a and b are ? its not an ordered pair ?


roger

 

[dextercioby]

What ordered pair (actually triplets) are u talking about??Is it relevant in any way?

Daniel.

 

[roger]

What ordered pair (actually triplets) are u talking about??Is it relevant in any way?

I refer to the coordinates

Daniel.

Well, I have no idea what the question asks or how to tackle it .

Please would you care to enlighten me ?


thankyou


Roger

 

[ankh]

I thought A and B are just two points. So, the line between them is B-A (8-1, 12+2,-3-4) You get (7,14,-7) Than you divide this by seven because there are seven parts total (ratio 2:5). You get (1,2,-1) --- This is for one part. So the answer is (1, -2, 4) + 2(1,2,-1) = (3,2,2)
I am not sure though, I didn't check my work

 

[BobG]

ankh hit it right on.

A=... is similar to an ordered pair, except in three dimensions instead of two (ordered 'triplets' if you will, although normally they're just called coordinates)

Same with B= ...

Each is a point. You have a vector that runs from point A to point B (that's what they mean by AB, a line segment running from A to B).

ankh hit on exactly how you find the vector that runs from A to B. Subtract x-coordinates from x-coordinates, y-coordinates from y-coordinates, etc.

Multiply the vector by 2/7 (this is scalar multiplication, so each component, x, y, and z, is multiplied by 2/7).

Add the result to your point of origin, or point A.

 

[roger]

thanx bob .

i've got one further query :

if this problem was in 2 dimensions example :

A=( 3,7) b= ( 7,15)

and it asks to find the coordinates of the point which divides AB by 2: 5, do you know how it could be done ?


thanks in advance.

Roger

 

[BobG]

Exactly the same way as for the three dimensional problem, except it's even easier. You only have x and y components to worry about instead of x, y, and z.

This is no different than what you learned in grade school. It just seems different. In grade school, they taught one dimensional vectors. Now they're just expanding it into new dimensions.

Actually, that's an exaggeration. There are some big differences, like three different kinds of multiplication, but the scalar multiplication you need for these problems is just like one dimensional multiplication.

The key is to perform your one dimensional operations separately for each dimension.

Your two dimensional problem has one other twist. The coordinates of your resulting point will be fractions instead of integers.

 

[HallsofIvy]

The point with such "coordinates", whether there are 2, 3, or 10000 is that you can treat each coordinate separately.

If you have A=(3,7) and B=(7,15) and want to find a point that divides AB in a "2:5 ratio", look at the x-coordinates, 3 and 7, first. You want to find a number that divides the interval from 3 to 7 in a "2:5 ratio", That means that you want to find a number that is 2 "intervals" from 3" and 5 "intervals" from 7. What are "intervals"? Whatever size fits. The distance from 3 to 7 is 7-3= 4 and that has to be divided into 2+5= 7 intervals: each interval has length 4/7. 2 intervals would have length 2*(4/7)= 8/7 so 2 intervals from 3 is 3+ 8/7= 29/7. 5 intervals would have length 5(4/7)= 20/7. Notice that 7- 20/7= 49/7- 20/7= 29/7 again. If we define an "interval" as 4/7, start at 3 and go up or start at 7 and go down, we get the same thing: 29/7.
29/7 divides the interval from 3 to 7 into 2 parts, in the ratio of 2:5. This is sometimes called a "weighted" average. If our point divides 3 to 7 in a "2:5 ratio", we clearly want it closer to 3 than to 7. That means, in calculating the value, we need to give 3 more "weight" than 7. Do that by multiplying 3 by the "5" of "2:5" and multiplying 7 by the "2". 5(3)+ 2(7)= 15+ 14= 29. Now divide by 2+5= 7 so that this is an "average": 29/7 as before.

Now do the same with the y-coordinate: we want a y value that will divide the interval from 7 to 15 in a "2:5 ratio". Okay 5(2)+ 2(15)= 10+ 30= 40 and we divide by 2+5= 7: the y coordinate is 40/7.

The point that divides the interval from (3,7) to (7,15) into a "2:5 ratio" is
(29/7, 40/7).

(Although it is not neccesary to do it, if you were to calculate the distance from (3,7) to (7,15), you would find that (29/7, 40/7) divides the distance into a "2:5 ratio" also.

Going back to your original question, "find a point that divides the interval from A = (1, - 2,4) and B = (8,12, - 3) into a 2:5 ratio", do exactly the same thing:

x-coordinates are 1 and 8. 5(1)+ 2(8)= 5+ 16= 21. 21/7= 3.
y-coordinates are -2 and 4. 5(-2)+ 2(12)= -10+ 24= 14. 14/7= 2.
Z-coordinates are 4 and -3. 5(4)+ 2(-3)= 20- 6= 14. 14/7= 2.

The point that divides the interval from (1, - 2,4) to (8,12, - 3) in a "2:5 ratio" is
(3, 2, 2).

Notice that the order of the points is important. Saying that the point divides the interval from (1, - 2,4) to (8,12, - 3) in a "2:5 ratio" means the point must be close to (1, -2, 4) (because 2 is less than 5).

If the problem said "divide the interval from (8,12,-3) to (1,-2,4) into a "2:5 ratio", we would calculate:
5(8)+ 2(1)= 40+ 2= 42. 42/7= 6.
5(12)+ 2(-2)= 60- 4= 56. 56/7= 8.
5(-3)+ 2(4)= -15+ 9= 7. 7/7= 1.

The point that divides the interal from (8,12,-3) to (1,-2,4) into a "2:5 ratio" is
(6, 8, 1).

If you were really good at drawing in 3-dimensional plots, you could mark those original two points, draw the straight line between them, divide it into 7 equal parts and find that the two points calculated are each 2 of those parts from the two end points and 5 parts from the other.

 

[roger]

The point with such "coordinates", whether there are 2, 3, or 10000 is that you can treat each coordinate separately.

If you have A=(3,7) and B=(7,15) and want to find a point that divides AB in a "2:5 ratio", look at the x-coordinates, 3 and 7, first. You want to find a number that divides the interval from 3 to 7 in a "2:5 ratio", That means that you want to find a number that is 2 "intervals" from 3" and 5 "intervals" from 7. What are "intervals"? Whatever size fits. The distance from 3 to 7 is 7-3= 4 and that has to be divided into 2+5= 7 intervals: each interval has length 4/7. 2 intervals would have length 2*(4/7)= 8/7 so 2 intervals from 3 is 3+ 8/7= 29/7. 5 intervals would have length 5(4/7)= 20/7. Notice that 7- 20/7= 49/7- 20/7= 29/7 again. If we define an "interval" as 4/7, start at 3 and go up or start at 7 and go down, we get the same thing: 29/7.
29/7 divides the interval from 3 to 7 into 2 parts, in the ratio of 2:5. This is sometimes called a "weighted" average. If our point divides 3 to 7 in a "2:5 ratio", we clearly want it closer to 3 than to 7. That means, in calculating the value, we need to give 3 more "weight" than 7. Do that by multiplying 3 by the "5" of "2:5" and multiplying 7 by the "2". 5(3)+ 2(7)= 15+ 14= 29. Now divide by 2+5= 7 so that this is an "average": 29/7 as before.

Now do the same with the y-coordinate: we want a y value that will divide the interval from 7 to 15 in a "2:5 ratio". Okay 5(2)+ 2(15)= 10+ 30= 40 and we divide by 2+5= 7: the y coordinate is 40/7.

The point that divides the interval from (3,7) to (7,15) into a "2:5 ratio" is
(29/7, 40/7).

(Although it is not neccesary to do it, if you were to calculate the distance from (3,7) to (7,15), you would find that (29/7, 40/7) divides the distance into a "2:5 ratio" also.

Going back to your original question, "find a point that divides the interval from A = (1, - 2,4) and B = (8,12, - 3) into a 2:5 ratio", do exactly the same thing:

x-coordinates are 1 and 8. 5(1)+ 2(8)= 5+ 16= 21. 21/7= 3.
y-coordinates are -2 and 4. 5(-2)+ 2(12)= -10+ 24= 14. 14/7= 2.
Z-coordinates are 4 and -3. 5(4)+ 2(-3)= 20- 6= 14. 14/7= 2.

The point that divides the interval from (1, - 2,4) to (8,12, - 3) in a "2:5 ratio" is
(3, 2, 2).

Notice that the order of the points is important. Saying that the point divides the interval from (1, - 2,4) to (8,12, - 3) in a "2:5 ratio" means the point must be close to (1, -2, 4) (because 2 is less than 5).

If the problem said "divide the interval from (8,12,-3) to (1,-2,4) into a "2:5 ratio", we would calculate:
5(8)+ 2(1)= 40+ 2= 42. 42/7= 6.
5(12)+ 2(-2)= 60- 4= 56. 56/7= 8.
5(-3)+ 2(4)= -15+ 9= 7. 7/7= 1.

The point that divides the interal from (8,12,-3) to (1,-2,4) into a "2:5 ratio" is
(6, 8, 1).

If you were really good at drawing in 3-dimensional plots, you could mark those original two points, draw the straight line between them, divide it into 7 equal parts and find that the two points calculated are each 2 of those parts from the two end points and 5 parts from the other.

Hallsoivy,

Theres two things i don't understand...

1.) : 5(8)+ 2(1)= 40+ 2= 42. 42/7= 6. using this as an example, I don't understand, why you multiply the 8 by 5 and 1 by 2 ?

2.) I don't fully understand why the order makes a difference, and why you can treat the x,y,z coordinates separately ?


Thanks


Roger

 

[HallsofIvy]

Hallsoivy,

Theres two things i don't understand...

1.) : 5(8)+ 2(1)= 40+ 2= 42. 42/7= 6. using this as an example, I don't understand, why you multiply the 8 by 5 and 1 by 2 ?

2.) I don't fully understand why the order makes a difference, and why you can treat the x,y,z coordinates separately ?


Thanks


Roger

Do you understand what "divide in a 2:5 ratio" means? To take a simple example, The points 0 and 7, on a number line, are exactly 7 units apart. The number "2" divides that interval in a 2:5 ratio- 2 is exactly 2 units from 0 and 5 units from 7. The number "5", on the other hand, divides that interval in a 5:2 ratio- it is 5 units from 0 and 2 units from 7.
to say that point P "divides the interval from A to B in a 2:5 ratio" means that it lies on the straight line between them, that the distance from A to P is 2/7 of the entire length, and that the distance from P to B is 5/7 of the entire length.
We "multiply the 8 by 5 and the 1 by 2" (rather than the other way around) for precisely the reason I gave before: In order to divide the interval from A to B into a "2:5" ratio, the point must be closer to A than to B. The point we want must closer to "8" than it is to "1" so we must "weight" the 8 more than the 1.

Again, look at that easy example: To find the point that divides the interval from 0 to 7 in a 2:5 ratio, I need to use a weighted average that gives the 0 point (since it should be closer) more weight: instead of multiplying each by 1/2 (as in (a+b)/2) which would be equal weights, I multiply by 5/7 and 2/7. The denominator "7" is, of course, 5+ 2, so that if we were looking for a "weighted average" of the SAME VALUES, a and a, we would have (5a+ 5a)/7= 7a/7= a.

As for why we can deal with the x, y, z, coordinates separately, well that's the whole point of a coordinate system with axes at right angles! You could use similar triangles to give a precise proof.

 

[Spiro09]

Look up the division point theorem:


OP= (b/a+b)OA + (a/a+b)OB