Question: How to quickly solve a system of equations where the unknowns are the coordinates of a vector?

With Maple, is there a way to quickly solve systems of equations where the solutions are the coordinates of a vector?
An example of an idea: A problem to solve:
There are points: M(1,0,0) and N(0,0-1); plane (Q): x-y-4=0. Determine the plane (P) containing M and N, making with (Q) a 45 degree angle.
Assume: Equation of (P): Ax+By+Cz+D=0
We have:
Normal vector of (P): np(vector):=<A,B,C>
Normal vector of (Q): nq(vector):=<1,-1,0>
Direction vector of line NM: dNM(vector):=<1,0,1>
To determine: A, B, C, it is necessary to solve the system of equations:
cos(45 degrees)= np(vector).nq(vector)/abs(np(vector)) x abs(nq(vector))=1/Sqrt(2)

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