n[{a_,b_,c_,d_}] := a^2+b^2+c^2+d^2;

x[{a_,b_,c_,d_},{e_,f_,g_,h_}] := {a e-b f-c g-d h,+b e+a f-d g+c h,  
+c e+a g+d f-b h,+b g-c f+d e+a h};  

conj[{a_,b_,c_,d_}]:= {a,-b,-c,-d};

inv[{a_,b_,c_,d_}] := 1/n[{a,b,c,d}]*conj[{a,b,c,d}];


p[{a_,b_,c_,d_}] := Sqrt[n[{a,b,c,d}]]/Sqrt[b^2+c^2+d^2] {b,c,d};

t = { 10,1,1,1};

w[r_] := {0, Sin[10 r], Cos[10 r] , 2 r     };


r[{a_,b_,c_,d_},{t_,xx_,y_,z_}] := x[x[{a,b,c,d},{t,xx,y,z}],inv[{a,b,c,d}]];


(* Equation of the plane perpendicular to (a,b,c,d) containing the (t,x,y,z).*)

check[{a_,b_,c_,d_},{x_,y_,z_}] := (

q1 = {x,y,z};

rhs = b x + c y + d z; 

(* Find piercing point *) 

t = rhs / (b^2 + c^2  + d^2); 

p0 = {t b, t c, t d}; 

q2 = p[r[{a,b,c,d},{0,x,y,z}]];

costheta =  (q1-p0).(q2-p0)/ (Sqrt[(q1-p0).(q1-p0)] Sqrt[(q2-p0).(q2-p0)]);

theta = ArcCos[costheta]);

f[tt_,{xx_,yy_,zz_}] := p[x[x[  tt ,{ 0,xx,yy,zz}],inv[tt]]];

look[t_] := Graphics3D[
                       {RGBColor[1,0,0], Line[{    
                              f[t,{ 0.1, 0.1, 0.1}],
                              f[t,{ 4.1, 0.1, 0.1}], 
                              f[t,{ 4.1, 0.1, 8.1}],
                              f[t,{ 4.1, 0.1, 0.1}],
                              f[t,{ 4.1, 4.1, 0.1}],
                              f[t,{ 4.1, 4.1, 8.1}],
                              f[t,{ 4.1, 4.1, 0.1}],
                              f[t,{ 0.1, 4.1, 0.1}],
                              f[t,{ 0.1, 4.1, 8.1}],
                              f[t,{ 0.1, 4.1, 0.1}],
                              f[t,{ 0.1, 0.1, 0.1}],
                              f[t,{ 0.1, 0.1, 8.1}],
                              f[t,{ 0.1, 4.1, 8.1}],
                              f[t,{ 4.1, 4.1, 8.1}],
                              f[t,{ 4.1, 0.1, 8.1}],
                              f[t,{ 0.1, 0.1, 8.1}]
                             }
                            ]
                       }
                      ];



v1 = look[ {  0 ,   1,   0,   0}  ];


Show[v1];