How to convert direction vector to euler angles?

Let's see if I understand correctly. This is about the orientation of a rigid body in three dimensional space, like an air plane during flight. The nose of that airplane points towards the direction vector

D=(XD,YD,ZD) .

Towards the roof is the up vector

U=(XU,YU,ZU) .

Then heading H would be the direction vector D projected onto the earth surface:

H=(XD,YD,0) ,

with an associated angle

angle_H=atan2(YD,XD) .

Pitch P would be the up/down angle of the nose with respect to the horizon, if the direction vector D is normalized you get it from

ZD=sin(angle_P)

resulting in

angle_P=asin(ZD) .

Finally, for the bank angle we consider the direction of the wings, assuming the wings are perpendicular to the body. If the plane flies straight towards D, the wings point perpendicular to D and parallel to the earth surface:

W0 = ( -YD, XD, 0 )

This would be a bank angle of 0. The expected Up Vector would be perpendicular to W0 and perpendicular to D

U0 = W0 × D

with × denoting the cross product. U equals U0 if the bank angle is zero, otherwise the angle between U and U0 is the bank angle angle_B, which can be calculated from

cos(angle_B) = Dot(U0,U) / abs(U0) / abs(U)
sin(angle_B) = Dot(W0,U) / abs(W0) / abs(U) .

Here 'abs' calculates the length of the vector. From that you get the bank angle as

angle_B = atan2( Dot(W0,U) / abs(W0), Dot(U0,U) / abs(U0) ) .

The normalization factors cancel each other if U and D are normalized.

we need three vectors: X1, Y1, Z1 of local coordinate system (LCS) expressed in terms of world coordinate system (WCS). The code below presents how to calculate three Euler angles based on these 3 vectors.

#include <math.h> 
#include <float.h> 

#define PI 3.141592653589793 
/**
 * @param X1x
 * @param X1y
 * @param X1z X1 vector coordinates
 * @param Y1x
 * @param Y1y
 * @param Y1z Y1 vector coordinates
 * @param Z1x
 * @param Z1y
 * @param Z1z Z1 vector coordinates
 * @param pre precession rotation
 * @param nut nutation rotation
 * @param rot intrinsic rotation
 */
void lcs2Euler(
        double X1x, double X1y, double X1z,
        double Y1x, double Y1y, double Y1z,
        double Z1x, double Z1y, double Z1z,
        double *pre, double *nut, double *rot) {
    double Z1xy = sqrt(Z1x * Z1x + Z1y * Z1y);
    if (Z1xy > DBL_EPSILON) {
        *pre = atan2(Y1x * Z1y - Y1y*Z1x, X1x * Z1y - X1y * Z1x);
        *nut = atan2(Z1xy, Z1z);
        *rot = -atan2(-Z1x, Z1y);
    }
    else {
        *pre = 0.;
        *nut = (Z1z > 0.) ? 0. : PI;
        *rot = -atan2(X1y, X1x);
    }
}