mM = 
      Transpose[{{Cos[1], Cos[2], Cos[3]}, {Sin[1], Sin[2], 
            Sin[3]}}]

Transpose[mM] . {(79/10), (54/10), -(9/10)}

FullSimplify[Inverse[Transpose[mM] . mM]]

mp = 
      FullSimplify[
        Inverse[Transpose[mM] . 
              mM] . (Transpose[mM] . {(79/10), 
                (54/10), -(9/10)})]

Plot[mp . {Cos[x], Sin[x]}, {x, 0, 2  Pi}, 
      Prolog -> {{Green, 
            PointSize[
              0.02], Point[#] & /@ {{1, (79/10)}, {2, 
                  (54/10)}, {3, -(9/10)}}}}]