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

Question

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?

Answer 1

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 $<1,0,-1>$

Answer 2

No need to guess. The cross product of the two given vectors isn’t going to work since it’s by definition orthogonal to both of them.

The solutions to the homogeneous linear equation $\mathbf v\cdot\mathbf x=0$ are all of the vectors $\mathbf x$ that are orthogonal to $\mathbf v$. I expect that you know how to find the general solution to such an equation. It shouldn’t be too hard from there to pick values for the free variables that produce a vector that isn’t orthogonal to the other vector, which you can check by computing a dot product.