Why are $v=(1,-2,2)$ and $w=(-1,0,3)$ not parallel? When are two vectors parallel?

Question

Why are $v=(1,-2,2)$ and $w=(-1,0,3)$ not parallel? When are two vectors parallel? Please help me. I can't understand this.

Answer 1

if they are parallel, then we have a real number $\alpha$ for which we have $$(1;-2;2)=\alpha(-1;0;3)$$ so we get $$1=\alpha$$ $$-2=0$$ $$2=3\alpha$$ which can not be

Answer 2

When are two vectors parallel?

If you have the coordinates, it's very easy to check: two vectors are parallel if they are (non-zero) scalar multiples of each other.

Why are $v=(1,-2,2)$ and $w=(-1,0,3)$ not parallel?

Following above; because there is no scalar $k$ such that $(1,-2,2)=k\cdot (-1,0,3)$.

---

Addition after the comment.

The vectors $(-6,0,4)$ and $(3,0,-2)$ are parallel because $(-6,0,4)= \color{blue}{-2} \cdot (3,0,-2)$.

Answer 3

They are not parallel because they are not multiple one of the other (i.e. they are linearly independent) that is we can't find any $k \in \mathbb{R}$ such that

$$\vec v=k \vec w$$

As an example $\vec v=(1,-2,2)$ and $\vec u=(-2,4,-4)$ are parallel with $k=-2$.