Find angle of line which runs perpendicular to a tangent trajectory of a point on 2d cartesian frame

Question

As below, I am looking to find the variable '$\theta_e$'. Assuming the parameters which are given are the point ($x,y$) , ($L_h$),($d_e$), v, ($\theta_d$), ($x_d,y_d$). I do not have the parameter '$\theta$'.

The only information I have about the problem is that the line '$d_e$' runs perpendicular to a tangent trajectory at ($x_d,y_d$) with an angle '$\theta_e$' .

Is it possible that I can find '$\theta_e$' ? Any idea?

Answer 1

If $Q$ is the common endpoint between segments $L_h$ and $d_e$, then we can compute its coordinates in two ways: $$ Q=(x+L_h\cos\theta,y+L_h\sin\theta)=(x_d+d_e\sin\theta_d,y_d-d_e\cos\theta_d). $$ From that we can then compute angle $\theta$ and finally: $\theta_e=\theta_d-\theta$.