Vector addition and parallel lines, from Chapter 1, Introduction to Linear Algebra - Strang

Question

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 $c\cdot v$".

-Thank you

Answer 1

For $(3)$ imagine fixing some integer $c$ and therefore, some vector $(c,0)$. This vector you can visualize as a point on the $x$-axis. If you add to this vector all vectors of the form $(0,d)$ where $d$ is some real number, you obtain the set $\{(c,d)\mid d \in \mathbb{R} \}$. All of these vectors you can visualize as points lying on the vertical line through $(c,0)$.

Once you see this, for $(4)$ imagine letting $c$ be any non-negative real number. Now instead of vertical lines passing through the $x$-axis at integer intervals, you have a vertical line for all non-negative real numbers. You can visualize this as the right half of the $xy$ plane.