How to move vector(points) coordinates, for a certain angle

I tried several things, but none worked as it should. How to move vector(points) coordinates, for a certain angle that I calculated!

I calculated $\beta = 88.7^\circ$.
I'd like to expand/skew vectors $DA$ and $DC$, to get perpendicular vectors($90^\circ$)(in my case for $1.3^\circ$). How to calculate that (I need to get coordinates).

After that I need to "move" the rectangle, to be perpendicular to the coordinate system (which is in the picture case the blue/black rectangle)

I don't need the result, I'd like to know how this is calculated.

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you can use the graham-schmidt process to change the basis to an orthogonal one. the linear transformation that does this is calculated by finding the coordinates of the new basis in terms of the old one. –  lyj Feb 1 '13 at 10:22

The computations required here are lengths.

The only things that matter here are the lengths $|DA|$, $|DC|$ and what happens to the point $D$.

So assuming that the point $D$ moves to $D'$, then the new coordinates will be $D'$, $C'=D'+(|DC|,0)$, $A'=D'+(0,-|DA|)$, $B'=D'+(|DC|,-|DA|)$.

Explicitly: If $A'=(x_{A'},y_{A'})$, $B'=(x_{B'},y_{B'})$, $C'=(x_{C'},y_{C'})$, $D'=(x_{D'},y_{D'})$, then

$x_{A'}=x_{D'},y_{A'}=y_{D'}-|DA|$,

$x_{B'}=x_{D'}+|DC|,y_{B'}=y_{D'}-|DA|$,

$x_{C'}=x_{D'}+|DC|,y_{C'}=y_{D'}$,

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the D point can stay on the same position, it doesnt matter. –  anzes Feb 1 '13 at 10:52
Then take $D'=D$. –  copper.hat Feb 1 '13 at 10:54
How to calculate for example: C ′ =D ′ +(|DC|,0). Like Calculating vector(Xa-Xb),(Ya-Yb)? –  anzes Feb 1 '13 at 11:37
If $C'=(x_{C'},y_{C'})$, then $x_{C'}= x_{D'}+|DC|$, $y_{C'}= y_{D'}$. –  copper.hat Feb 1 '13 at 11:42
are this move formulas suitable only for perpendicular rectangle or have I just derived the wrong formulas for A' and B'? –  anzes Feb 1 '13 at 12:37

If you started with a rectangle, you'd first shear with $k = \cos \beta$ and then rotate it with $\sin = \frac{DC_Y}{DC}$ and $\cos = \frac{DC_X}{DC}$. As you are interested in the reverse procedure, simply take the inverse of this matrix. Note, however that this is not a unique transformation, satisfying your constraints.

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