# Help to find a solution

• June 24th 2014, 12:52 PM
Haytham1111
Help to find a solution
A triangle has corners P = (1,0), Q = (7,0) and R = ((7)^(1/2))-5, 2). We rotate the triangle 2.3 radians around the point R. What are the coordinates of the vertices of the new triangle? Also, find the matrix that performs this rotation in a homogen coordinates?

can you help me to solve this problem.... Thank you
• June 24th 2014, 04:22 PM
HallsofIvy
Re: Help to find a solution
One way to do this:
1) translate so that R is the origin by subtracting R from each vector.
P becomes $(1- (7^{1/2}- 5), 0- 2)= (6- \sqrt{7}, -2)$ and Q becomes $(7- (7^{1/2}- 5),-2)= (12- \sqrt{7}, -2)$
2) Rotate through by 2.3 radians by multiplying those vectors by the matrix $\begin{pmatrix}cos(2.3) & -sin(2.3) \\ sin(2.3) & cos(2.3)\end{pmatrix}$.
3) Translate back by adding R to each vector.

As for "homogeneous coordinates" do you know what they are? We represent the point (x, y) as the three-vector (x, y, 1) with the understanding that if, after some transformation, the last component is not 1, we divide through by it. That is, (x, y, a) represents the same point as (x/a, y/a, 1). In homogeneous coordinates, a translation of (x, y) by (a, b) is given by the matrix multiplication $\begin{pmatrix}1 & 0 & a \\ 0 & 1 & b \\ 0 & 0 & 1\end{pmatrix}\begin{pmatrix}x \\ y \\ 1\end{pmatrix}= \begin{pmatrix}x+ a \\ y+ b\\ 1\end{pmatrix}$ while rotation through angle $\theta$ is given by the matrix multiplication $\begin{pmatrix}cos(\theta) & -sin(\theta) & 0 \\ sin(\theta) & cos(\theta) & 0 \\ 0 & 0 & 1\end{pmatrix}\begin{pmatrix}x \\ y \\ 1\end{pmatrix}= \begin{pmatrix}x cos(\theta)- y sin(\theta) \\ x sin(\theta)+ y cos(\theta) \\ 1\end{pmatrix}$.

Find the matrices corresponding to the translation of R to the origin, rotation by 2.3 radians, and translation back and multiply them (in the correct order).
• June 25th 2014, 12:59 AM
Haytham1111
Re: Help to find a solution
Hei, tanks for replying. I tried to continue the prosess in har first part and i multiples These vectors to the rotation matrices and Then added R to them, but i did not git the right answer. Can you show how we can do this

The same for the homogenous coordinates. What would be x , y , a and b in this question