$(x,y,1)$ is perpendicular to $(1,2,-7)$. What is the relationship between $x$ and $y$?

Question

$(x,y,1)$ is perpendicular to $(1,2,-7)$. What is the relationship between $x$ and $y$?

I think that there is no relationship between $x$ and $y$ because they could be any number. I know that the dot product of both $x$ and $y$ must be $0$ but that hasn't helped me get anywhere.

Answer 1

Given two non-null vectors $u,v$, they are orthogonal ("perpendicular") if and only if $$u\cdot v=0.$$

So given $(x,y,1)$ and $(1,2,-7)$, that gives us the equation: $$(x,y,1)\cdot (1,2,-7) = x + 2y -7=0$$ So the relationship between $x$ and $y$ is $$\boxed{x=7-2y}$$