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Further maths issues

I'm struggling with these two questions 4 & 6,

The matrix is 3X3 meaning it's a 3D transformation, but in the question they are all 2x2 matrices, so I don't understand?

Also I can't get an equation for the locus of Q in question 6.

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3FDB4B35-79BD-463F-910D-14CB71731C8C.jpg.jpeg

Reply 1

Joe_fish217

Oh for 6.a) I got u = x^2 - y^2
V = 2xy

What unit is this ?

Im sitting WJEC new spec , but I'm
Not sure where this exact question is from

Reply 4

Pangol

Original post by Joe_fish217

The matrix is 3X3 meaning it's a 3D transformation, but in the question they are all 2x2 matrices, so I don't understand?

Instead of using 2x2 matrices operating on 2x1 column vectors, you need to use 3x3 matrices operating on 3x1 column vectors.

For any usual 2x2 transformation matrix, use the 3x3 version with the final row and column consisting of zeros, except for the bottom right entry which is a 1. For any 2x1 colum vector, use the 3x1 column vector which is the same, but with a 1 in the extra space.

The advantage of this is that if you use the matrix;

1 0 a
0 1 b
0 0 1

you get a translation by a in the x-direction and b in the y-direction. You can represent translations in the normal 2x2 way, as the origin is always invariant.

I would also be interested in where this question has come from - I've not seen this at A level before.

Reply 5

Joe_fish217

Yeah I used that but it didn't give me the matrix I should get , I used

(edited 6 years ago)

Attachment not found

And this is for AS further pure maths WJEC, but the practice paper came from a further maths support group website

Reply 8

Pangol

Original post by Joe_fish217

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Your middle matrix should be

1 0 2
0 1 1
0 0 1

Reply 9

Pangol

Original post by Joe_fish217

And this is for AS further pure maths WJEC, but the practice paper came from a further maths support group website

The new WJEC spec does include matrices that represent translations, so it looks like you might be expected to do this. It's a new one one me!

Reply 10

Joe_fish217

Why is the middle matrix that? I was shown to look what happens to point 1,0 and 0,1 and that gives you the matrix?

Reply 11

Pangol

Original post by Joe_fish217

Why is the middle matrix that? I was shown to look what happens to point 1,0 and 0,1 and that gives you the matrix?

This is true for transformations that leave the origin unchanged, so it's fine for stretches, enlargements, rotations and shears. But it won't work for translations, which is why we need to introduce a "dummy" third row/column. All of the "action" that goes on in a translation needs to be contained in this new third column. It is a bit of a fudge, but it needs to be to include translations in a maxtric scheme.

You can do the same thing with translations in 3D, but then you need to setp up from 3x3 to 4x4 matrices!

Reply 12

Joe_fish217

Okay I'll just look up this "dummy" column method for translations, thanks , any advice on the complex transformation question?

Reply 13

Pangol

Original post by Joe_fish217

any advice on the complex transformation question?

Let me first say that I'm not too familiar with this kind of question, so I'm rather making this up as I go along. But try this;

- use the locus of P to get an equation of the form y^2 = (something in terms of x)
- you can then use this to write u in terms of x only.
- rearrange this to get x^2 in terms of u.
- square the expression you have for v. In this expression, replace y^2 by the expression you have in terms of x from the first step.
- now you should have v^2 entirely in terms of x, or more usefully, in terms of x^2. Replace every instance of x^2 by the expression in terms of u that you have from the third step.

At the end of all of this, you have an equation in u and v only. It does start v^2 = ... instead of v = ..., but I don't think that's a problem.