Let $x_n=a_n+b_{n}i$, where $a_n,b_n\in \mathbb{R}$, and $n\in \mathbb{N}$, and suppose that the sequence $x_n$ is governed by the recurrence relation $x_{n+2}=x_n+im{x}_{n+1}$, where $m\in \mathbb{R}$ . We investigate the conditions under which two subsequences $x_{2k}$ and $x_{2k+1}$ lie on perpendicular lines in the complex plane. Specifically, we express $x_1$ and $x_2$ as $x_1=a_1+b_{1}i$ and $x_2=a_2+b_{2}i$, satisfying the condition $a_1{a}_2+b_1{b}_2=0$.

We show that there exist two perpendicular lines $l_1$ and $l_2$ passing through the origin in the complex plane, such that for all $k\in \mathbb{N}$, the sequence alternates between the lines: $x_{2k}\in l_1$ and $x_{2k+1}\in l_2$. This provides a geometric interpretation of the recurrence relation. Furthermore, the ratios $\frac{b_{2k}}{a_{2k}}=\frac{b_2}{a_2}$ and $\frac{b_{2k+1}}{a_{2k+1}}=\frac{b_1}{a_1}$remain constant for all $k\in \mathbb{N}$.