in complex plane we have a line passing through $z_1$ and $z_2$. I want to find a line making an angle $\theta$ with this line and passing through $z_1$. How do I do this?
I know i can convert it to Cartesian by splitting it into real and imaginary parts and applying the rotation matrix. But I feel it might be easier in doing it directly in complex plane since rotation there is only a multiplication by $e^{i\theta}$ or $e^{-i\theta}$ depending on anticlockwise or clockwise direction.
Can someone please help me solve this completely and get the final expression? Regards
Best Answer
One line is the set of points $\{z_1+t(z_2-z_1)e^{i\theta}: t \in \mathbb R\}$ and the other is $\{z_1+t(z_2-z_1)e^{-i\theta}:t \in \mathbb R\}$. [Since the original line is along $z_2-z_1$ you only have to rotate this vector].
You can also write the equation to the first line as $\frac {z-z_1} {|z-z_1|}=\frac {{(z_2-z_1)}e^{i\theta}} {|z_2-z_1|}$.