Triangles in the Gauss Plane What is the image of $z$ ($z=a+ib$) in $C$ ($z$ is complex number) under reflection in the line which passes through $O$ at angle $\theta$ to the real axis?
Since one rotation can be factored into a product of two reflections, which means the question contains 3 reflections? Anyone has hint? What should I do?
The slope of the specified line is $\tan\theta$ or $\frac{\sin\theta}{\cos\theta}$. The slope of any normal to the specified line is the negative reciprocal of that (or $- \frac{\cos\theta}{\sin\theta}$). Plug $a$, $b$ and that value into the point-slope equation of a line. Solve that equation and the equation of the original line $\left(\sin\theta\right)x=\left(\cos\theta\right)y$ simultaneously to find the intersection of the two (say, $(c|d)$. The coördinates of the desired reflection point are $(2c-a|2d-b)$ or: $- a + 2 a \cos^2 θ + 2 b \cos θ \sin θ + (- b + 2 a \cos θ \sinθ + 2 b \sin^2 θ) i$.
The slope of the specified line is $\tan\theta$ or $\frac{\sin\theta}{\cos\theta}$. The slope of any normal to the specified line is the negative reciprocal of that (or $- \frac{\cos\theta}{\sin\theta}$). Plug $a$, $b$ and that value into the point-slope equation of a line. Solve that equation and the equation of the original line $\left(\sin\theta\right)x=\left(\cos\theta\right)y$ simultaneously to find the intersection of the two (say, $(c|d)$. The coördinates of the desired reflection point are $(2c-a|2d-b)$ or: $- a + 2 a \cos^2 θ + 2 b \cos θ \sin θ + (- b + 2 a \cos θ \sinθ + 2 b \sin^2 θ) i$.