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Au101

Member

Registered: 2010-12-01

Posts: 353

Matrices - Area Scale Factor

Hi guys, I have another matrices question.

The rectangle R has vertices at the points (-1,0),(0,-3),(4,0) and (3,3).

The matrix

Where a is a constant

a) Find, in terms of a, the coordinates of the vertices of the image of R under the transformation given by A.

By simple matrix multiplication, we can find that the answer to this is: (2,-1), (3a-9,-3a), (-8,4) and (3-3a,3a+3)

b) Find det(A), leaving your answer in terms of a

This is

Given that the area of the image of R is 75

c) Find the positive value of a.

In fairness, I'm not sure if I've done the modulus right, it's been a while since I did that . Even so, the answer should be 2, and I don't see how to get there

Does anyone have any ideas?

Thanks,
Au101

Last edited by Au101 (2010-12-19 09:38:38)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Matrices - Area Scale Factor

Hi Au101;

For the area of the rectangle R, I am getting 15, what is wrong?

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: Matrices - Area Scale Factor

Hi Bobby

Thanks, 15 sounds like exactly the right answer, maybe I'm just tired and am doing something silly, but when I drew the rectangle I got a 5 x √10 rectangle. I assume it should be 5 x 3, but surely if the coordinates of our two points are (-1,0) and (0,-3) we need to do pythagoras, or have I constructed the rectangle incorrectly. It's entirely possible I have .

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Matrices - Area Scale Factor

I am getting two triangles of b = 5 and h = 3 so that should equal A = 15.

As a further problem down the road. I do not get a = 2. I am getting a = -8.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: Matrices - Area Scale Factor

Hmmm that's a good point I get a = 8 from A = 15, too, but I'm still getting A = 5√10, here's what I did

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Matrices - Area Scale Factor

Hi;

A = (b*h)/2

That still equals 15. The left triangle on the bottom is (1*3)/2 = 3 / 2.
The other triangle on the bottom is (3 * 4 ) / 2 = 6. Total area below the x axis is 6 + 3 / 2 = 15 / 2. The area above is the same. A = 15.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: Matrices - Area Scale Factor

Hmmmm...That's interesting, but surely the area of a rectangle is given by the width multiplied by the height.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Matrices - Area Scale Factor

Hi;

If you look at my drawing you will see that it is not a rectangle. The angles are not right angles.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: Matrices - Area Scale Factor

Hmmm, how very curious, given that the question explicitly states

'The rectangle R has vertices at the points (-1,0),(0,-3),(4,0) and (3,3).'

I must say I am thoroughly confused Thanks for all of your help, once again!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Matrices - Area Scale Factor

Hi Au101;

Where did the question come from, a book, or your teacher. If from a book, can you give me the title?

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: Matrices - Area Scale Factor

Certainly, it is the newest edition of:

Edexcel AS and A Level Modular Mathematics FP1 (Further Pure Mathematics 1). This is the textbook accompanying the Advanced Level examinations set by the examining body Edexcel in Britain.

The ISBN number is: 978-0-435519-23-0 and is distributed by Heinemann.

See:

http://www.pearsonschoolsandfecolleges.co.uk/Secondary/Scotland/Maths/EdexcelModularMathematicsforASandALevel/ISBN/FurtherPureMathematics/EdexcelASandALevelModularMathematicsFurtherPureMathematics1.aspx (http://tinyurl.com/34dgwdm)

Last edited by Au101 (2010-12-19 11:58:04)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Matrices - Area Scale Factor

Hi;

The rectangle R has vertices at the points (-1,0),(0,-3),(4,0) and (3,3).

In my opinion it is a typo. That is not a rectangle.

The slope of AD is -3 and the slope of CD is 3 / 4. In order for those two lines to be perpendicular they must have slopes that are negative reciprocals. -3 and 3 / 4 are not negative reciprocals.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: Matrices - Area Scale Factor

Hmmm you're quite right, thanks Bobbym.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Matrices - Area Scale Factor

Hi;

If you scale your graph ( x ticks and y ticks the same length ) something that lots of programs do not do right you can see it visually.

There is tons of mathematical evidence but I thought you would like to see it.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: Matrices - Area Scale Factor

Thanks Bobbym for the trouble. You are absolutely right, I've decided to move on. I have to say, I really don't think much of this textbook, but what can you do

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Matrices - Area Scale Factor

I have been to their site to ask for addenda to the book. But I cannot find any way to contact them. Mistakes do creep into textbooks but ...

I found it, you can go here.

http://www.pearsonschoolsandfecolleges.co.uk/Home.aspx

There is a contact us. Send them an email with the problem # and the book title.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: Matrices - Area Scale Factor

Oooh, well found, thanks!

Bob

Administrator

Registered: 2010-06-20

Posts: 10,649

Re: Matrices - Area Scale Factor

hi Au101 and bobbym

I agree it is not a rectangle and its area is = 15

I agree with your det = - a - 3

If the object has area = 75, that makes the scale factor x5.

|-a -3 | = a + 3

so a + 3 = 5  => a = 2

In fairness, I'm not sure if I've done the modulus right, it's been a while since I did that . Even so, the answer should be 2, and I don't see how to get there

So it looks like the question is ok apart from the word 'rectangle'.

See diagram.  Using a = 2, shows both parallelograms, with areas of 15 and 75 respectively.

If you letter the points ABCD and the corresponding points A'B'C'D' you will find that A to B to C to D has opposite sense to A' to B' to C' to D';  ie if one is clockwise, the other is anticlockwise.  This explains why the scale factor is strictly negative but converted by |Det|.
Bob

ps.  I taught this course a long time ago but no longer have the text book.  Post again if you need any more assistance.

Last edited by Bob (2010-12-20 02:40:54)

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Au101

Member

Registered: 2010-12-01

Posts: 353

Re: Matrices - Area Scale Factor

Ah, thanks bob, so it would seem that I did indeed make a mistake with the modulus signs - thanks!