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Suggested coordinate system for linear detector elements in a plane perpendicular to z.
Assume line direction is chosen to go in the positive $y$ direction. If the line is parallel to the x-axis, choose the positive direction to be the positive x direction. Then angle of inclination of the line is an angle $\phi$, 0 <= phi < 180. Unit vector along the line is lhat = xhat cos phi + yhat sin phi, the unit vector normal to the line (in the detector plane) is nhat = -xhat sin phi + yhat cos phi. In this way, lhat and nhat satisfy the right hand rule, i. e., lhat cross nhat = zhat.