This question was taken from MIT OCW and Introduction to Linear Algebra by Gilbert Strang
Write the dot product of $(1,4,5)$ and $(x,y,z)$ as a matrix multiplication of $A\overrightarrow { x } $. The matrix $A$ has one row. The solutions to $A\overrightarrow { x } = 0$ lie on a ____ perpendicular to the vector ____. The columns of $A$ are only in ____-dimensional space.
What I understand thus far and what I need clarification on:
1) The solutions to $A\overrightarrow { x } = 0$ lie on a plane. However, the answer saids: "...on a plane in three dimensions". Why?
2) ...perpendicular to the vector ____. I don't understand this part. The solution to this is the $0$ vector, no?
3) The columns of $A$ are only in one-dimensional space, since matrix $A$ is $[1,4,5]$