Thread: Transforation geometry -- using matrices!

This is a question from GCSE level, so please offer a simple explanation if possible.

I have been given the following:

To rotate a point (x,y) 90deg anti-clockwise, pre-multiply the x y matrix with:

Code:

( 0 -1 )
  1  0

Similarly, for a an anti-clockwise rotation of 180deg, it's:

Code:

-1  0
0   -1

p.s., how can I create a matrix in this forum in the math tag?

It's tough to remember so memorize so many matrices when there are very similar looking matrices for reflection (on x axis, y axis and y = -x). I think perhaps, an explanation of how these matrices are created will help me. Or maybe some pointers on how to memorize at least 8-9 such matrices.

Thanks ...

Originally Posted by struck

This is a question from GCSE level, so please offer a simple explanation if possible.

I have been given the following:

To rotate a point (x,y) 90deg anti-clockwise, pre-multiply the x y matrix with:

Code:

( 0 -1 )
  1  0

Similarly, for a an anti-clockwise rotation of 180deg, it's:

Code:

-1  0
0   -1

p.s., how can I create a matrix in this forum in the math tag?

It's tough to remember so memorize so many matrices when there are very similar looking matrices for reflection (on x axis, y axis and y = -x). I think perhaps, an explanation of how these matrices are created will help me. Or maybe some pointers on how to memorize at least 8-9 such matrices.

Thanks ...

The matrix that rotates by an anti-clockwise angle of is




p.s. The latex code I used for generating this matrix is

[tex]\left( \begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta \end{array} \right)[/tex]

I have found plenty to help me with counter clockwise rotations but when its clockwise the values i get for theta when i do arc cos and arc sin are both different, can you tell me why please?

Max

Originally Posted by max

I have found plenty to help me with counter clockwise rotations but when its clockwise the values i get for theta when i do arc cos and arc sin are both different, can you tell me why please?

Max

Give an example.

To rotate clockwise, all you do is replace with in the rotation matrix.

Find theta and hence the angle of rotation and the direction of

top row of matrix (-1/√2) (1/√2)
bottom row (-1/√2) (-1/√2)

Clockwise direction but
when you calculate arccos (-1/√2) you get 135 deg which is correct but
arcsin (-1/√2) is -45 deg

What am I missing?

Max

matrix attached

Originally Posted by max

Find theta and hence the angle of rotation and the direction of

top row of matrix (-1/√2) (1/√2)
bottom row (-1/√2) (-1/√2)

Clockwise direction but
when you calculate arccos (-1/√2) you get 135 deg which is correct but
arcsin (-1/√2) is -45 deg

What am I missing?

Max

matrix attached

and are both negative in the third quadrant.

So or .

Thank you for your reply and I understand where the -135 deg comes from. Its negative due to being a clockwise rotation about the origin (anti clockwise it is 225 deg as you say), where I am getting lost is why do I get -45 deg for calculating arcsin (-1/√2) ?

Originally Posted by max

Thank you for your reply and I understand where the -135 deg comes from. Its negative due to being a clockwise rotation about the origin (anti clockwise it is 225 deg as you say), where I am getting lost is why do I get -45 deg for calculating arcsin (-1/√2) ?

That's the correct answer, but to the wrong question!

The correct question is:

What value(s) of simultaneously satisfy

.... (1)

.... (2)

Ahh okay thank you that actually makes it so much easier to understand.

Max