The discussion centers on solving a vector system involving sum and dot product equations, specifically focusing on the relationship between vectors ##\vec x##, ##\vec y##, and constants ##\vec u##, ##\vec v##, and ##m##. The key solution presented is ##\vec x = \vec u - \vec y##, with the understanding that if ##\vec v = \langle 0,0,1\rangle## and ##m=0##, then ##\vec x## can represent any vector in the ##xy## plane. The conversation emphasizes the need for clarity in problem specifications, particularly regarding dimensions and known variables, to derive accurate solutions.
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You didn't specify whether this is a 2D or 3D problem. You also didn't specify that ##\vec u, ~ \vec v,~ m## are given constants and the unknowns are ##\vec x## and ##\vec y##. Is that correct? Certainly, if it is a 3D problem there can be many solutions. For example if ##\vec v = \langle 0,0,1\rangle## and ##m=0##, so ##\vec x## is perpendicular to ##\vec v## so ##\vec x## can be any vector in the ##xy## plane. Then no matter what ##\vec u## is ##\vec y = \vec u - \vec x## is a solution. So ##\vec u## and ##\vec x ## can be all over the place.LCSphysicist said:
Unless there was more info provided, I see no way to guess this 'solution' is what was wanted. Indeed, I would propose a better 'solution' is to omit ##\vec b## and define a as a unit vector. ##\frac{m\vec v}{|\vec v|^2}+\lambda\hat a##, where ##\vec v.\hat a=0##, generates all solutions of ##\vec x.\vec v=m##.LCSphysicist said: