old exam 366t105.pdf: 2 & 3
text problem: 4.3

#2: let b1=.95, and g1 the related gamma (particle as seen in S')
U'=g1 c (0,b1,0,i)

let b2=.99, and g2 the realted gamma (S' velocity as seen in S)
Note to boost from S'->S use the inverse transformation or a beta=-b2

<<LorentzBoost.m                                                        
up= g1 {0,-b1,0,I}                                                       
b1=.95                                                                  
g1=1/Sqrt[1-b1^2]                                                       
u=boost[-.99,0,0].up
Out[5]= {22.4753 + 0. I, -3.04243 + 0. I, 0. + 0. I, 0. + 22.7024 I}
v={I u[[1]]/u[[4]],I u[[2]]/u[[4]]}                                    
v=Re[v]                                                                
Norm[v]                                                                
Out[10]= 0.999029
ArcTan[v[[2]]/v[[1]]]*180./Pi                                          
Out[11]= 7.70914

#3
P^2=a^2(3^2+2^2+1^2-4^2)=-2 a^2 =-(mc)^2; m=sqrt(2)a/c
4 a= gamma mc = gamma sqrt(2)a
4/sqrt(2)=gamma
V/c= (3,-2,1)/4; speed=sqrt(3^2+2^2+1^2)/4 c = sqrt(7/8) c

check: does that beta give hte above gamma=1/sqrt(1-7/8)=sqrt(8) (check)

4.3
I'll cheat and use Mathematica
a={{1,2,4},{-1,0,3},{5,-5,3}}
b={{-4,4},{1,5},{3,2}}
In[8]:= a.b                                                                     

Out[8]= {{10, 22}, {13, 2}, {-16, 1}}
In[9]:= Transpose[a]                                                            

Out[9]= {{1, -1, 5}, {2, 0, -5}, {4, 3, 3}}

In[10]:= Transpose[b]                                                           

Out[10]= {{-4, 1, 3}, {4, 5, 2}}

In[11]:= Transpose[%8]                                                          

Out[11]= {{10, 13, -16}, {22, 2, 1}}

In[12]:= %10.%9                                                                 

Out[12]= {{10, 13, -16}, {22, 2, 1}}