Solve this vector system containing sum and dot product equations

[LCSphysicist]

Homework Statement

All below

Relevant Equations

All below

Seems to me the answer is a specific vector:

The second forms a plane, while the first X is just a vector. The intersection between the λX that generates the (properties of all vectors that lie in the...) plane (i am not saying X is the director vector!)

How to write this in vector language?

 

[FactChecker]

What are you supposed to solve for? The first equation gives $\vec x$ precisely: $\vec x = \vec u - \vec y$. What more is there to do?

 

[LCKurtz]

Homework Statement:: All below
Relevant Equations:: All below

View attachment 266376

Seems to me the answer is a specific vector:

The second forms a plane, while the first X is just a vector. The intersection between the λX that generates the (properties of all vectors that lie in the...) plane (i am not saying X is the director vector!)

How to write this in vector language?

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]

The problem is not i didn't specify, the question is really just it.
Seeing the answer, it write x and y in terms of others vectors a and b.
The answers is right to me, but at same time a little biased, as you both said.

 

[haruspex]

The problem is not i didn't specify, the question is really just it.
Seeing the answer, it write x and y in terms of others vectors a and b.
The answers is right to me, but at same time a little biased, as you both said.

View attachment 266391

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 $\vec x.\vec v=m$.