I'm having a little bit of trouble with understanding one of the worked example from Gilbert Strang's Introduction to Linear Algebra book section 1.1, question 1.1B. It goes as follows (I've attached the image as well).
Question 1.1 B
For $v = (1,0)$ and $w = (0,1)$, describe all points $c\cdot v$ with
(1) whole numbers $c$
(2) non-negative $c\geq0$.
(3) Then add all vectors $d\cdot w$ and
(4) describe all $c\cdot v + d\cdot w$.
And the answer given was,
(1) The vectors $c\cdot v = (c,0)$ with whole numbers $c$ are equally spaced points along the $x-$axis (the direction of $v$). They include $(-2,0)$, $(- 1,0)$, $(0,0)$, $(1,0)$, $(2,0)$.
(2) The vectors $c\cdot v$ with $c\geq0$ fill a half-line. It is the positive $x-$axis. This half-line starts at $(0,0)$ where $c = 0$. It includes $(\pi,0)$ but not $(-\pi,0)$.
(3) Adding all vectors $d\cdot w = (0,d)$ puts a vertical line through those points $c\cdot v$. We have infinitely many parallel lines from (whole number $c$, any number $d$).
(4) Adding all vectors $d\cdot w$ puts a vertical line through every $c\cdot v$ on the half-line. Now we have a half-plane. It is the right half of the $xy$-plane (any $x\geq0$, any height $y$).
Could anyone please explain the 3rd and 4th answers? I don't see how it can make parallel lines. Also, I don't understand what it means by "Adding all vectors $d\cdot w$ puts a vertical line through every $c\cdot v$".