Let $x$ be a non-zero vector, orthogonal to vectors $y + z$ and $z$, with $x, y, z \in \mathbb R^2$. Prove that $y$, $y - z$ and $z - y$ are orthogonal to $x$ and parallel to $z$.
To prove they are orthogonal I tried just by dot product, but how to do with the parallel question?
In $\mathbb R^2$, two vectors that are orthogonal to the same vector must be parallel to each other. This is how you get the "parallel to $z$" conclusion of the problem.
In higher dimensions, the above is not true. Subsequently, the "parallel" part fails. An example could be given in $\mathbb R^3$ by letting $y,z$ be horizontal and $x$ vertical.