Transform XY plane entities to an arbitrary plane

Question

I have a profile entity in XY plane. I have to do an extrusion along a vector. So I have to bring the profile entity perpendicular to the extrusion vector. Extrusion vector can be a arbitrary one (Xi+Yj+Zk). Could you please provide some idea about vector base manipulation to bring the profile perpendicular to the extrusion vector?

Update

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Profile in XY plane

Global up vector : (0,0,1)

Extrusion vector : (Xe,Ye,Ze)

Local right = Extrusion vector x Global up vector = (x1,y1,z1) Local Left = Extrusion vector x Local right = (x2,y2,z2)

I assume I have to rotate the plane twice about Local right and Local Left.

So the angle is calculated for Local right = Atan(Xe.Y/Xe.Z) and the entity is rotated about Local right by calculated angle. and it is working fine

My question is how to calculate the angle for Local Left ?

If I summarize, I have to do the opposite of this one (Mapping coordinates from plane given by normal vector to XY plane)

Answer 1

If you have your extrusion vector and one reference vector you can cross multiply them to retrieve your 2D planes' right vector in 3D space.

I usually use the global up vector as my reference. So your extrusion vector cross multiplied with global up will result in a local right vector. Another cross multiply between your extrusion vector and your new local right vector will result in a local up vector.

Once you get both your local right and local up vectors, those will be your planes X and Y vectors in 3D space.