Q:

Find a vector orthogonal to another vector $x$ but not vector $y$?

So I know how to find cross product, but our professor never showed us a problem like this where a vector CANNOT be orthogonal to another vector.

Here is the question plus my guess
Did I do this right at all? Or is there another way to tackle this?

A:

It is true that $<-2,2,-2>$ is orthogonal to $<1,2,1>$, but it is also orthogonal to $<1,0,-1>$ as you can observe with dot products.  So this is not quite a good answer
Since $<1,0,-1>$ is also orthogonal to $<1,2,1>$, you could try something like $<-2,2,-2> + <1,0,-1> \;=\; <-1,2,-3>$.  This will be orthogonal to $<1,2,1>$ but not to $<1,0,-1>$

