class TVector3: public TObject

The Physics Vector package 
-*                    ==========================                       
-* The Physics Vector package consists of five classes:                
-*   - TVector2                                                        
-*   - TVector3                                                        
-*   - TRotation                                                       
-*   - TLorentzVector                                                  
-*   - TLorentzRotation                                                
-* It is a combination of CLHEPs Vector package written by             
-* Leif Lonnblad, Andreas Nilsson and Evgueni Tcherniaev               
-* and a ROOT package written by Pasha Murat.                          
-* for CLHEP see:  http://wwwinfo.cern.ch/asd/lhc++/clhep/             
*

TVector3

TVector3 is a general three vector class, which can be used for the description of different vectors in 3D.

Declaration / Access to the components

TVector3 has been implemented as a vector of three Double_t variables, representing the cartesian coordinates. By default all components are initialized to zero:

  TVector3 v1;        // v1 = (0,0,0)
  TVector3 v3(1,2,3); // v3 = (1,2,3)
  TVector3 v4(v2);    // v4 = v2

It is also possible (but not recommended) to initialize a TVector3 with a Double_t or Float_t C array.

You can get the basic components either by name or by index using operator():

  xx = v1.X();    or    xx = v1(0);
  yy = v1.Y();          yy = v1(1);
  zz = v1.Z();          zz = v1(2);

The memberfunctions SetX(), SetY(), SetZ() and SetXYZ() allow to set the components:

  v1.SetX(1.); v1.SetY(2.); v1.SetZ(3.);
  v1.SetXYZ(1.,2.,3.);
 

Noncartesian coordinates

To get information on the TVector3 in spherical (rho,phi,theta) or cylindrical (z,r,theta) coordinates, the
the member functions Mag() (=magnitude=rho in spherical coordinates), Mag2(), Theta(), CosTheta(), Phi(), Perp() (the transverse component=r in cylindrical coordinates), Perp2() can be used:

  Double_t m  = v.Mag();    // get magnitude (=rho=Sqrt(x*x+y*y+z*z)))
  Double_t m2 = v.Mag2();   // get magnitude squared
  Double_t t  = v.Theta();  // get polar angle
  Double_t ct = v.CosTheta();// get cos of theta
  Double_t p  = v.Phi();    // get azimuth angle
  Double_t pp = v.Perp();   // get transverse component
  Double_t pp2= v.Perp2();  // get transvers component squared

It is also possible to get the transverse component with respect to another vector:

  Double_t ppv1 = v.Perp(v1);
  Double_t pp2v1 = v.Perp2(v1);

The pseudo-rapidity ( eta=-ln (tan (theta/2)) ) can be obtained by Eta() or PseudoRapidity():
 
  Double_t eta = v.PseudoRapidity();

There are set functions to change one of the noncartesian coordinates:

  v.SetTheta(.5); // keeping rho and phi
  v.SetPhi(.8);   // keeping rho and theta
  v.SetMag(10.);  // keeping theta and phi
  v.SetPerp(3.);  // keeping z and phi
 

Arithmetic / Comparison

The TVector3 class provides the operators to add, subtract, scale and compare vectors:

  v3  = -v1;
  v1  = v2+v3;
  v1 += v3;
  v1  = v1 - v3
  v1 -= v3;
  v1 *= 10;
  v1  = 5*v2;

  if(v1==v2) {...}
  if(v1!=v2) {...}
 

Related Vectors

  v2 = v1.Unit();       // get unit vector parallel to v1
  v2 = v1.Orthogonal(); // get vector orthogonal to v1

Scalar and vector products

  s = v1.Dot(v2);   // scalar product
  s = v1 * v2;      // scalar product
  v = v1.Cross(v2); // vector product

 Angle between two vectors

  Double_t a = v1.Angle(v2);

Rotations

Rotation around axes

  v.RotateX(.5);
  v.RotateY(TMath::Pi());
  v.RotateZ(angle);

Rotation around a vector

  v1.Rotate(TMath::Pi()/4, v2); // rotation around v2

Rotation by TRotation

TVector3 objects can be rotated by objects of the TRotation class using the Transform() member functions,
the operator *= or the operator * of the TRotation class:

  TRotation m;
  ...
  v1.transform(m);
  v1 = m*v1;
  v1 *= m; // Attention v1 = m*v1

Transformation from rotated frame

  TVector3 direction = v.Unit()
  v1.RotateUz(direction); // direction must be TVector3 of unit length

transforms v1 from the rotated frame (z' parallel to direction, x' in the theta plane and y' in the xy plane as well as perpendicular to the theta plane) to the (x,y,z) frame.