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I have an expression:
{((-2*m*n + B*R0^2)*Sin[t])/(4*Pi*R0), -1/4*((-2*m*n + B*R0^2)*Cos[t])/(Pi*R0), 0}
And I would like it to recognize this pattern
%/.{-Sin[t],Cos[t],0}-> ET
such that
{((-2*m*n + B*R0^2)*Sin[t])/(4*Pi*R0), -1/4*((-2*m*n + B*R0^2)*Cos[t])/(Pi*R0), 0} /.{-Sin[t],Cos[t],0}-> ET
should result in
-ET (-2*m*n + B*R0^2)/(4*Pi*R0)
or something the like. It however does not work. Who can help?
I think your result has a typo. Instead of -ET (2*m*n + B*R0^2)/(4*Pi*R0) it should read -ET (-2*m*n + B*R0^2)/(4*Pi*R0), otherwise there is no solution.
A further problem is the minus sign in front of the Cos term. MMA always uses the full form of an expression for matching. Look at:
((-2 m n + B R0^2) Cos[t])/(4 \[Pi] R0) // FullForm
(* Times[Rational[1,4],Power[Pi,-1],Power[R0,-1],Plus[Times[-2,m,n],Times[B,Power[R0,2]]],Cos[t]] *)
and
-(((-2 m n + B R0^2) Cos[t])/(4 \[Pi] R0)) // FullForm
(* Times[Rational[-1,4],Power[Pi,-1],Power[R0,-1],Plus[Times[-2,m,n],Times[B,Power[R0,2]]],Cos[t]] *)
You see, the minus is integrated in the Rational[-1,4]. Therefore, for matching, the minus can not be matched separately. Here is how you can do it:
{((-2*m*n + B*R0^2)*Sin[t])/(4*Pi*R0), -1/4*((-2*m*n + B*R0^2)*Cos[t])/(Pi*R0), 0} /. { x__ Sin[t], y__ Cos[t], 0} :> ET y
Consider vector projection using the dot product. Let v be the input vector and p be the pattern:
v = {((-2*m*n + B*R0^2)*Sin[t])/(4*Pi*R0), -1/4*((-2*m*n + B*R0^2)*Cos[t])/(Pi*R0), 0};
p = {-Sin[t], Cos[t], 0};
Then the parts of the vector which are in line with the given pattern are:
Simplify[v.p/Norm[p], Element[t, Reals]]
(2 m n - B R0^2)/(4 π R0)
You may multiply this by ET, if you wish. For other expression the Simplify may not be necessary to achieve a nice result, and I have assumed t is a real number here for simplicity. If that is not necessarily the case, this method may not be appropriate.
Note also that the non-conforming parts of the vector are given by:
Simplify[v-v.p/Norm[p], Element[t, Reals]]
{0, 0, 0}
In general two vectors will not be perfectly aligned with each other, so this term is normally non-zero. This is the part of your vector which does not co-align with your given pattern.
