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Math Help - Transofrmation

  1. #1
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    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?
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  2. #2
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    Hello Keep
    Quote Originally Posted by Keep View Post
    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 z-axis is perpendicular to the x-y plane, and passes through the origin. So a rotation along the z-axis is simply a rotation in the x-y 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 45^o is: \begin{pmatrix} \dfrac{1}{\sqrt2} & -\dfrac{1}{\sqrt2}\\\dfrac{1}{\sqrt2} & \dfrac{1}{\sqrt2}\end{pmatrix}

    The translation is described by adding the column matrix \binom{1}{3} to the column vectors representing the original points; i.e.

    \binom00 + \binom13 = \binom13,\quad \binom21 + \binom13 = \binom34, \quad\binom12 + \binom13 = \binom25

    2 I presume the scaling has centre the origin, in which case the matrix you need is: \begin{pmatrix} \frac{1}{2} & 0\\0 & \frac{1}{2}\end{pmatrix}

    Can you complete this now?

    Grandad
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  3. #3
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    Quote Originally Posted by Grandad View Post
    Hello Keep1 The z-axis is perpendicular to the x-y plane, and passes through the origin. So a rotation along the z-axis is simply a rotation in the x-y 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 45^o is: \begin{pmatrix} \dfrac{1}{\sqrt2} & -\dfrac{1}{\sqrt2}\\\dfrac{1}{\sqrt2} & \dfrac{1}{\sqrt2}\end{pmatrix}

    The translation is described by adding the column matrix \binom{1}{3} to the column vectors representing the original points; i.e.

    \binom00 + \binom13 = \binom13,\quad \binom21 + \binom13 = \binom34, \quad\binom12 + \binom13 = \binom25

    2 I presume the scaling has centre the origin, in which case the matrix you need is: \begin{pmatrix} \frac{1}{2} & 0\\0 & \frac{1}{2}\end{pmatrix}

    Can you complete this now?

    Grandad
    I first translated it by (1,3) so that wil give the points \binom13,\quad\binom34,\quad\binom25,\quad

    Now rotating this along the z-axis by 45 degrees will give \begin{pmatrix} {-1.4} & -0.7 & -2.1\\2.8 & {4.9}&{4.9}\end{pmatrix}

    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
    \begin{pmatrix} {-0.7} & -0.35 & -1.05\\1.4 & {2.45}&{2.45}\end{pmatrix}?

    Thank you!
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