How to rotate a vector around a particular axis

Question

The .X3D format has an interesting rotation system. Unlike most formats containing rotation values around the X, Y and Z axis, .X3D gives a normalized direction vector and then gives a value in radians for rotation around that axis.

Example:

The axis to rotate around: 0.000000 0.465391 0.885105 
Rotation around that axis (in radians): 3.141593

I have the conversion from radians to degrees, but I need the rotation values around XYZ from these values.

Answer 1

We can build a sequence for elementary matrix transformations to rotate about angle-axis. Let axis unit vector is w, angle Theta. Auxiliary values:

V = Sqrt(wx*wx + wz*wz)
W = Sqrt(wx*wx + wy*wy + wz*wz)  //1 for unit dir vector
Cos(Alpha) = wz/V
Sin(Alpha) = wx/V
Cos(Beta) = V/W
Sin(Beta) = wy/W

Transformation sequence:
Ry(-Alpha)   //rotation matrix about Y-axis by angle -Alpha
Rx(Beta)
Rz(Theta)
Rx(-Beta)
Ry(Alpha) 

Note that for is axis is parallel to Y , one should use usual just rotation matrix about Y (accounting for direction sign), because V value is zero.

---

There is rather complex Rodrigues' rotation formula for computing the rotation matrix corresponding to a rotation by an angle Theta about a fixed axis specified by the unit vector w.

Explicit matrix here (weird formatted picture):

Answer 2

This Below C++ Function Rotates a Point Around A Provided Center and Uses Another Vector(Unit) as Axis to rotate Around. This Code Below Was Made Thanks to Glenn Murray who provided the Formula Provided in Link. This is Tested and is Working Perfectly As intended. Note: When Angle Is Positive And Unit Vector is {0,0,1} It Rotates To the Right Side, Basically Rotation Is On the Right Side, Axes are {x-forward,y-right,z-up}

void RotateVectorAroundPointAndAxis(Vector3D YourPoint, Vector3D PreferedCenter, Vector3D UnitDirection, float Angle, Vector3D& ReturnVector)
{
    float SinVal = sin(Angle * 0.01745329251);
    float CosVal = cos(Angle * 0.01745329251);
    float OneMinSin = 1.0f - SinVal;
    float OneMinCos = 1.0f - CosVal;   

    UnitDirection = GetUnitVector(UnitDirection, { 0,0,0 });// This Function Gets unit Vector From InputVector - DesiredCenter

    float Temp = (UnitDirection.x * YourPoint.x) + (UnitDirection.y * YourPoint.y) + (UnitDirection.z * YourPoint.z);
    ReturnVector.x = (PreferedCenter.x * (UnitDirection.y * UnitDirection.y)) - (UnitDirection.x * (((-PreferedCenter.y * UnitDirection.y) + (-PreferedCenter.z * UnitDirection.z)) - Temp));
    ReturnVector.y = (PreferedCenter.y * (UnitDirection.x * UnitDirection.x)) - (UnitDirection.y * (((-PreferedCenter.x * UnitDirection.x) + (-PreferedCenter.z * UnitDirection.z)) - Temp));
    ReturnVector.z = (PreferedCenter.z * (UnitDirection.x * UnitDirection.x)) - (UnitDirection.z * (((-PreferedCenter.x * UnitDirection.x) + (-PreferedCenter.y * UnitDirection.y)) - Temp));

    ReturnVector.x = (ReturnVector.x * OneMinCos) + (YourPoint.x * CosVal);
    ReturnVector.y = (ReturnVector.y * OneMinCos) + (YourPoint.y * CosVal);
    ReturnVector.z = (ReturnVector.z * OneMinCos) + (YourPoint.z * CosVal);

    ReturnVector.x += (-(PreferedCenter.z * UnitDirection.y) + (PreferedCenter.y * UnitDirection.z) - (UnitDirection.z * YourPoint.y) + (UnitDirection.y * YourPoint.z)) * SinVal;    
    ReturnVector.y += ( (PreferedCenter.z * UnitDirection.x) - (PreferedCenter.x * UnitDirection.z) + (UnitDirection.z * YourPoint.x) - (UnitDirection.x * YourPoint.z)) * SinVal;
    ReturnVector.z += (-(PreferedCenter.y * UnitDirection.x) + (PreferedCenter.x * UnitDirection.y) - (UnitDirection.y * YourPoint.x) + (UnitDirection.x * YourPoint.y)) * SinVal;    
}