Given two vectors
[tex]v=(a,b),\quad w=(c,d)[/tex]
they are parallel if they are a multiple of each other:
[tex]c=ka,\quad d=kb[/tex]
You can easily test this by checking if
[tex]\dfrac{c}{a}=\dfrac{d}{b}[/tex]
they are orthogonal if their dot product is null:
[tex]v\cdot w=ac+bd=0[/tex]
For example, in the first case, we have
[tex]\dfrac{10}{3}\div 2 \neq \dfrac{4}{3}\div 5[/tex]
So, they aren't parallel. Similarly, you have
[tex]2\cdot \dfrac{10}{3}+5\cdot \dfrac{4}{3}\neq 0[/tex]
So, they aren't orthogonal.
In the second case, we have
[tex]\dfrac{5}{-12}\neq \dfrac{-6}{-10}[/tex]
So, they aren't parallel, and
[tex]5(-12)+(-6)(-10)=-60+60=0[/tex]
So, they are orthogonal.
Finally, we have
[tex]\dfrac{2}{-4}=\dfrac{-7}{14}=-\dfrac{1}{2}[/tex]
So, they are parallel (and thus can't be orthogonal)
