Question

do vectors (1,2,3) and (3,6,9) make a line or a plane? and same for (1,0,0) and (0,2,3)?

Answer 1

The vectors (1,2,3) and (3,6,9) are multiples of each other so all the linear combinations of these form a line.

The vectors (1,0,0) and (0,2,3) are independent so they form a plane.

Answer 2

just solve the matrix and u will see it with any numbers \[\left[\begin{matrix}1 & 2 &3 \\ 3 & 6 & 9\end{matrix}\right]\] or \[\left[\begin{matrix}1 & 0 &0 \\ 0 & 2 & 3\end{matrix}\right]\] try to readuce ( get 0 0 0 verse) one verse with the other using the linear operation - it is the easiest way - it means that you can create one of thoese vectors with the others