s=
1
4
(
2
Abs[Sin[bs]+Cos[bs]
′
g
[t]]
+(
2
Abs[Cos[s]]
+
2
Abs[Sin[s]]
)
2
Abs[Cos[bs]-Sin[bs]
′
g
[t]]
)(2(2
2
a
+
2
t
+4
2
b
2
t
+4atCos[bs]+
2
t
Cos[2bs])Sin[bs]+((4
2
a
+(3+4
2
b
)
2
t
)Cos[bs]+t(4a(1+
2
b
+Cos[2bs])+tCos[3bs]))
′
g
[t]+2
2
g[t]
((1+2
2
b
-Cos[2bs])Sin[bs]+Cos[bs](1+4
2
b
-Cos[2bs])
′
g
[t])+4g[t](a(-1-
2
b
+Cos[2bs])+tCos[bs](-1+
2
b
+Cos[2bs])-(t-
2
b
t+2aCos[bs]+tCos[2bs])Sin[bs]
′
g
[t]))-(
2
Abs[btCos[bs]-bg[t]Sin[bs]]
+
2
Abs[-Cos[s](a+tCos[bs])+bCos[bs]g[t]Sin[s]+(Cos[s]g[t]+btSin[s])Sin[bs]]
+
2
Abs[bCos[s](Cos[bs]g[t]+tSin[bs])+Sin[s](a+tCos[bs]-g[t]Sin[bs])]
)(a+tCos[bs]-g[t]Sin[bs])
′′
g
[t]
In[]:=
vecN=(a*Cos[s]*Sin[b*s]+a*
′
g
[t]*Cos[s]*Cos[b*s]-t*b*Sin[s]+t*Cos[s]*Sin[b*s]*Cos[b*s]+t*
′
g
[t]*Cos[s]*
2
Cos[b*s]
-g[t]*
′
g
[t]*b*Sin[s]-g[t]*Cos[s]*
2
Sin[b*s]
-g[t]*
′
g
[t]*Cos[s]*Sin[b*s]*Cos[b*s])((1+
2
′
g
[t]
)*
2
a
+
2
t
*
2
b
+(1+
2
′
g
[t]
)*(
2
t
*
2
Cos[b*s]
)+
2
b
*
2
g[t]
*
2
′
g
[t]
+(1+
2
′
g
[t]
)*
2
g[t]
*
2
Sin[b*s]
+2a*t*
2
Sin[b*s]
*
2
Cos[b*s]
+(1+
2
′
g
[t]
)*(2a*t*Cos[b*s])-(1+
2
′
g
[t]
)*(2a*g[t]*Sin[b*s])+2t*
2
b
*g[t]*
′
g
[t]-(1+
2
′
g
[t]
)*(2t*g[t]*Sin[b*s]*Cos[b*s])),(a*Sin[s]*Sin[b*s]+a*
′
g
[t]*Sin[s]*Cos[b*s]+t*b*Cos[s]+t*Sin[s]*Sin[b*s]*Cos[b*s]+t*
′
g
[t]*Sin[s]*
2
Cos[b*s]
+b*g[t]*
′
g
[t]*Cos[s]-g[t]*Sin[s]*
2
Sin[b*s]
-g[t]*
′
g
[t]*Sin[s]*Sin[b*s]*Cos[b*s])((1+
2
′
g
[t]
)*
2
a
+
2
t
*
2
b
+(1+
2
′
g
[t]
)*(
2
t
*
2
Cos[b*s]
)+
2
b
*
2
g[t]
*
2
′
g
[t]
+(1+
2
′
g
[t]
)*
2
g[t]
*
2
Sin[b*s]
+2a*t*
2
Sin[b*s]
*
2
Cos[b*s]
+(1+
2
′
g
[t]
)*(2a*t*Cos[b*s])-(1+
2
′
g
[t]
)*(2a*g[t]*Sin[b*s])+2t*
2
b
*g[t]*
′
g
[t]-(1+
2
′
g
[t]
)*(2t*g[t]*Sin[b*s]*Cos[b*s])),(-a*Cos[b*s]+a*
′
g
[t]*Sin[b*s]-t*
2
Cos[b*s]
+t*
′
g
[t]*Sin[b*s]*Cos[b*s]+g[t]*Sin[b*s]*Cos[b*s]-g[t]*
′
g
[t]*
2
Sin[b*s]
)((1+
2
′
g
[t]
)*
2
a
+
2
t
*
2
b
+(1+
2
′
g
[t]
)*(
2
t
*
2
Cos[b*s]
)+
2
b
*
2
g[t]
*
2
′
g
[t]
+(1+
2
′
g
[t]
)*
2
g[t]
*
2
Sin[b*s]
+2a*t*
2
Sin[b*s]
*
2
Cos[b*s]
+(1+
2
′
g
[t]
)*(2a*t*Cos[b*s])-(1+
2
′
g
[t]
)*(2a*g[t]*Sin[b*s])+2t*
2
b
*g[t]*
′
g
[t]-(1+
2
′
g
[t]
)*(2t*g[t]*Sin[b*s]*Cos[b*s]));​​vecX={x1,x2,x3};​​scalar=s;​​prod=vecN.vecX;​​lsg=Solve[prodscalar,x1]//Flatten
Out[]=
x1(
2
b
2
t
+2at
2
Cos[bs]
2
Sin[bs]
+2
2
b
tg[t]
′
g
[t]+
2
b
2
g[t]
2
′
g
[t]
+
2
a
(1+
2
′
g
[t]
)+2atCos[bs](1+
2
′
g
[t]
)+
2
t
2
Cos[bs]
(1+
2
′
g
[t]
)-2ag[t]Sin[bs](1+
2
′
g
[t]
)-2tCos[bs]g[t]Sin[bs](1+
2
′
g
[t]
)+
2
g[t]
2
Sin[bs]
(1+
2
′
g
[t]
))s-(x2(btCos[s]+aSin[s]Sin[bs]+tCos[bs]Sin[s]Sin[bs]-g[t]Sin[s]
2
Sin[bs]
+bCos[s]g[t]
′
g
[t]+aCos[bs]Sin[s]
′
g
[t]+t
2
Cos[bs]
Sin[s]
′
g
[t]-Cos[bs]g[t]Sin[s]Sin[bs]
′
g
[t]))(
2
b
2
t
+2at
2
Cos[bs]
2
Sin[bs]
+2
2
b
tg[t]
′
g
[t]+
2
b
2
g[t]
2
′
g
[t]
+
2
a
(1+
2
′
g
[t]
)+2atCos[bs](1+
2
′
g
[t]
)+
2
t
2
Cos[bs]
(1+
2
′
g
[t]
)-2ag[t]Sin[bs](1+
2
′
g
[t]
)-2tCos[bs]g[t]Sin[bs](1+
2
′
g
[t]
)+
2
g[t]
2
Sin[bs]
(1+
2
′
g
[t]
))-(x3(-aCos[bs]-t
2
Cos[bs]
+Cos[bs]g[t]Sin[bs]+aSin[bs]
′
g
[t]+tCos[bs]Sin[bs]
′
g
[t]-g[t]
2
Sin[bs]
′
g
[t]))(
2
b
2
t
+2at
2
Cos[bs]
2
Sin[bs]
+2
2
b
tg[t]
′
g
[t]+
2
b
2
g[t]
2
′
g
[t]
+
2
a
(1+
2
′
g
[t]
)+2atCos[bs](1+
2
′
g
[t]
)+
2
t
2
Cos[bs]
(1+
2
′
g
[t]
)-2ag[t]Sin[bs](1+
2
′
g
[t]
)-2tCos[bs]g[t]Sin[bs](1+
2
′
g
[t]
)+
2
g[t]
2
Sin[bs]
(1+
2
′
g
[t]
))(-btSin[s]+aCos[s]Sin[bs]+tCos[s]Cos[bs]Sin[bs]-Cos[s]g[t]
2
Sin[bs]
+aCos[s]Cos[bs]
′
g
[t]+tCos[s]
2
Cos[bs]
′
g
[t]-bg[t]Sin[s]
′
g
[t]-Cos[s]Cos[bs]g[t]Sin[bs]
′
g
[t])
In[]:=
Simplify(
2
b
2
t
+2at
2
Cos[bs]
2
Sin[bs]
+2
2
b
tg[t]
′
g
[t]+
2
b
2
g[t]
2
′
g
[t]
+
2
a
(1+
2
′
g
[t]
)+2atCos[bs](1+
2
′
g
[t]
)+
2
t
2
Cos[bs]
(1+
2
′
g
[t]
)-2ag[t]Sin[bs](1+
2
′
g
[t]
)-2tCos[bs]g[t]Sin[bs](1+
2
′
g
[t]
)+
2
g[t]
2
Sin[bs]
(1+
2
′
g
[t]
))s-(x2(btCos[s]+aSin[s]Sin[bs]+tCos[bs]Sin[s]Sin[bs]-g[t]Sin[s]
2
Sin[bs]
+bCos[s]g[t]
′
g
[t]+aCos[bs]Sin[s]
′
g
[t]+t
2
Cos[bs]
Sin[s]
′
g
[t]-Cos[bs]g[t]Sin[s]Sin[bs]
′
g
[t]))(
2
b
2
t
+2at
2
Cos[bs]
2
Sin[bs]
+2
2
b
tg[t]
′
g
[t]+
2
b
2
g[t]
2
′
g
[t]
+
2
a
(1+
2
′
g
[t]
)+2atCos[bs](1+
2
′
g
[t]
)+
2
t
2
Cos[bs]
(1+
2
′
g
[t]
)-2ag[t]Sin[bs](1+
2
′
g
[t]
)-2tCos[bs]g[t]Sin[bs](1+
2
′
g
[t]
)+
2
g[t]
2
Sin[bs]
(1+
2
′
g
[t]
))-(x3(-aCos[bs]-t
2
Cos[bs]
+Cos[bs]g[t]Sin[bs]+aSin[bs]
′
g
[t]+tCos[bs]Sin[bs]
′
g
[t]-g[t]
2
Sin[bs]
′
g
[t]))(
2
b
2
t
+2at
2
Cos[bs]
2
Sin[bs]
+2
2
b
tg[t]
′
g
[t]+
2
b
2
g[t]
2
′
g
[t]
+
2
a
(1+
2
′
g
[t]
)+2atCos[bs](1+
2
′
g
[t]
)+
2
t
2
Cos[bs]
(1+
2
′
g
[t]
)-2ag[t]Sin[bs](1+
2
′
g
[t]
)-2tCos[bs]g[t]Sin[bs](1+
2
′
g
[t]
)+
2
g[t]
2
Sin[bs]
(1+
2
′
g
[t]
))(-btSin[s]+aCos[s]Sin[bs]+tCos[s]Cos[bs]Sin[bs]-Cos[s]g[t]
2
Sin[bs]
+aCos[s]Cos[bs]
′
g
[t]+tCos[s]
2
Cos[bs]
′
g
[t]-bg[t]Sin[s]
′
g
[t]-Cos[s]Cos[bs]g[t]Sin[bs]
′
g
[t])
Out[]=
2btx2Cos[s]-2ax3Cos[bs]-2tx3
2
Cos[bs]
+2ax2Sin[s]Sin[bs]-2x2g[t]Sin[s]
2
Sin[bs]
+x3g[t]Sin[2bs]+tx2Sin[s]Sin[2bs]+2bx2Cos[s]g[t]
′
g
[t]+2ax2Cos[bs]Sin[s]
′
g
[t]+2tx2
2
Cos[bs]
Sin[s]
′
g
[t]+2ax3Sin[bs]
′
g
[t]-2x3g[t]
2
Sin[bs]
′
g
[t]+tx3Sin[2bs]
′
g
[t]-x2g[t]Sin[s]Sin[2bs]
′
g
[t]-2s
2
a
+
2
b
2
t
+2atCos[bs]+
2
t
2
Cos[bs]
+
1
2
at
2
Sin[2bs]
+
2
(a+tCos[bs])
2
′
g
[t]
-g[t](2(a+tCos[bs])Sin[bs]-2
2
b
t
′
g
[t]+2(a+tCos[bs])Sin[bs]
2
′
g
[t]
)+
2
g[t]
(
2
Sin[bs]
+(
2
b
+
2
Sin[bs]
)
2
′
g
[t]
)(2btSin[s]-2aCos[s]Sin[bs]-tCos[s]Sin[2bs]-2Cos[s]Cos[bs](a+tCos[bs])
′
g
[t]+g[t](2Cos[s]
2
Sin[bs]
+(2bSin[s]+Cos[s]Sin[2bs])
′
g
[t]))