Math Help - Transofrmation

You are working on points in a 2D space. You have a triangle with the points (0,0),(2,1),(1,2). Now the question is: Give the following transformations as matrices and use these on the triangle:

1. Translation along (1,3) followed by a rotation along the z-axis by 45 degrees.
2. Scaling by 50% followed by a transformation of (2,1).

Here is my attempt but got stuck at the beginning.

The first one; a translation of (1,3) will give the points of the triangle as (1,3);(3,4);(2,5) now rotation along the z-axis, is that not a 3D space?

Hello Keep

Originally Posted by Keep

You are working on points in a 2D space. You have a triangle with the points (0,0),(2,1),(1,2). Now the question is: Give the following transformations as matrices and use these on the triangle:

1. Translation along (1,3) followed by a rotation along the z-axis by 45 degrees.
2. Scaling by 50% followed by a transformation of (2,1).

Here is my attempt but got stuck at the beginning.

The first one; a translation of (1,3) will give the points of the triangle as (1,3);(3,4);(2,5) now rotation along the z-axis, is that not a 3D space?

1 The -axis is perpendicular to the plane, and passes through the origin. So a rotation along the -axis is simply a rotation in the plane about the origin. Every point in the plane will be rotated on to another point in the plane, and so it is really a 2D transformation.

The matrix that will rotate about the origin through is:

The translation is described by adding the column matrix to the column vectors representing the original points; i.e.



2 I presume the scaling has centre the origin, in which case the matrix you need is:

Can you complete this now?

Grandad

Originally Posted by Grandad

Hello Keep1 The -axis is perpendicular to the plane, and passes through the origin. So a rotation along the -axis is simply a rotation in the plane about the origin. Every point in the plane will be rotated on to another point in the plane, and so it is really a 2D transformation.

The matrix that will rotate about the origin through is:

The translation is described by adding the column matrix to the column vectors representing the original points; i.e.



2 I presume the scaling has centre the origin, in which case the matrix you need is:

Can you complete this now?

Grandad

I first translated it by (1,3) so that wil give the points

Now rotating this along the z-axis by 45 degrees will give

I don't know whether the scaling is by the origin; we weren't given so I presume that it is by the origin. So scaling by 50% basically means dividing by two, right? Which will give ?

Thank you!