Linear Algebra - Geometric Application

Question

I need help with this question:

Find the values of $x$ such that $(x, 1-2x , 3)$ and $(1, -x, 3x)$ are parallel

Answer 1

In general, given two vectors $u$ and $v$, they are parallel iff $u = a v $ where $a \in \mathbb{F}$ (scalar). Thus, for your two vectors to be parallel, we want

$$ (x, 1 - 2x, 3) = a ( 1, -x, 3x ) $$

Which implies that

$$ x = a \; \; \; \text{or} \; \; \; x = \frac{1}{2-a} $$

as long as $a \neq 2$. Therefore, ....