Use MATLAB to solve the following problems related to rigid motion. Use different colors to display your results.

1. Simulate the rigid motion with matrix R and translation o_{\beta'} prescribed by Equation 4.4 in PRO-2014-Lecture.pdf. Basis \beta equals the standard basis \sigma.

% approximate rotation
R = [0.8047   -0.5059   -0.3106
     0.3106    0.8047   -0.5059
     0.5059    0.3106    0.8047];

% less approximate rotation
[U,D,V] = svd(R);
R = U*V';

% translation o_{\beta'}
o = [1;1;1];

1. Find the coordinates of vectors of \beta' in \beta and vice versa.
2. Plot vectors of \beta and \beta' in the standard basis, list the numeric values.
3. Plot coordinate systems (O=0,\beta) and (O',\beta'). i.e. plot the basic vectors as bound vectors originating from points O and O', respectively, list the numeric values.
4. Plot the bound vector X_\beta = [1;2;3] representing point X in (0,\sigma), list the numeric values.
5. Plot the position vector in (O',\beta') of point Y represented in (O',\beta') by vector Y_{\beta'} = [1;2;3], list the numeric values.
6. Consider point Z, where X moves by the motion given above. Plot the bound vector representing the point Z w.r.t. (O=0,\beta), list the numeric values.

Upload via the course ware the zip archive hw04.zip containing

1. hw04.pdf report file describing your solution with all figures
2. hw04.m MATLAB source code, which generates the results and figures for the report
3. all your additional MATLAB files required by hw04.m