Q:

Linear algebra - how to tell where vectors lie?

I'm working my way (self-study) through Strang's text on Linear Algebra and am currently on Problem 1.2 #6.
6b) The vectors that are perpendicular to $V = (1,1,1)$ lie on a _ . 
6c) The vectors that are perpendicular to $V = (1,1,1)$ and $(1,2,3)$ lie on a __.
The back of the book gives the answer as plane and line, respectively. I'm not sure I understand why or how to go about solving this. I know that two lines are perpendicular to each other if their dot product is $0$. I'm also familiar with the intuition behind this in $\mathbb R^2$, but would appreciate some help on how to generalize this to $\mathbb R^3$ and beyond.

A:

Any vector $(x,y,z)$ perpendicular to $(1,1,1)$ satisfies the condition $x+y+z=0$. The collection of such vectors is by definition a plane. 
If in addition the vector is perpendicular to $(1,2,3)$ then we have the added condition $x+2y+3z=0$. Together with $x+y+z=0$ this means all vectors of the type $(a,-2a,a)$ which is a line in the given plane. 

