Geometrical Proof of a Rotation

Question

I wanna prove geometrically ( and not by linear algebra, doing transformations in the bases ) the result of the rotation of a point. The proof should only include geometrical steps like using similarity between triangles, pythagorean theorem and definition of cos and sen on a triangle, for example.

This picture clears my doubt :

http://postimg.org/image/z60zv5d83/

Thanks in advance.

Answer 1

Let $(x,y)$ be the original coordinates, and $(x',y')$ the later one. Then $$x'=ax+by,~y'=cx+dy$$as they are linear combination. Notice that $$x^2+y^2=(x')^2+(y')^2$$Replacing $x',y'$ by $x,y$ in the above equation, we get $$a^2+c^2=b^2+d^2=1,~ab+cd=0$$Let $a=\cos\theta$ and do some work gives the result.

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As required by OP, I will give an alternative method using vector decomposition:

Let $\vec{OA}=(x,y)$ be the original vector, $\vec{OB}=(x',y')$ the vector after rotation of angle $\theta$. Then $$(x',y')=\vec{OB}=\vec{OC}+\vec{OD}=(x,y)\cos\theta+(y,-x)\sin\theta.~~\mbox{(DONE!)}$$

Answer 2

No transformation of bases is needed. Let $R:=R_\theta$ be the rotation about the origin by angle $\theta$.

All is needed, that this is a linear transformation. This can be viewed as a purely geometrical statement:

1. $0$ goes to $0$.
2. Rotation preserves line segments and takes triangles to triangles (corresponding to $R(a+b)=R(a)+R(b)$ for vectors $a,b$).
3. Rotation is exchangable with all zooms from the origin, i.e. $R(\lambda a)=\lambda R(a)$.

Let $e_1=(1,0)$ and $e_2=(0,1)$. Now, by definition of sine and cosine, we have that $R(e_1)=(\cos\theta,\sin\theta)$, and please check (geometrically, i.e. by drawing) that $R(e_2)=(-\sin\theta,\cos\theta)$.

Then, for our arbitrary point $P=(d,f)$, rotating it about $0$ is the same as rotating the $\vec{OP}$ vector, so by linearity, we have $$R(P)=R(d\cdot e_1+f\cdot e_2)=d\cdot R(e_1)+f\cdot R(e_2)\,,$$ and we're there.

Answer 3

Oh, i just thought of an easy solution :

http://postimg.org/image/o4lvhlg57/

But i just wanted some way to understand whats happening in order to be able to write the formulas steadily witouth memorizing.

The formula says that the point suffers a dcos(theta ) - fsen(theta) horizontal displacement for example. I wanted to know intuitively why thats true.