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: Take two pencils of approximately equal length. Hold them together at a right angle. Leave one of the pencils in a fixed position and try to figure out all the different ways you can move the second pencil to make a right angle with the first. You'll see that sweeps out a plane.
Hold the two pencils at a right angle. Take a third pencil of any length and make it perpendicular to the other two. You will see why it is a line.
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. 
A: (1,1,1) is a vector. So vectors perpendicular to it ll lie in a plane. To get a feel put your finger perpendicularly on the wall. Your finger is (1,1,1) and the wall is the plane. For the second bit think reverse. wall contains the two vectors and your finger is the new line perpendicular to it.
