Question

How do i figure out these problems ?

Answer 1

Given two vectors

$v=(a,b),\quad w=(c,d)$

they are parallel if they are a multiple of each other:

$c=ka,\quad d=kb$

You can easily test this by checking if

$\dfrac{c}{a}=\dfrac{d}{b}$

they are orthogonal if their dot product is null:

$v\cdot w=ac+bd=0$

For example, in the first case, we have

$\dfrac{10}{3}\div 2 \neq \dfrac{4}{3}\div 5$

So, they aren't parallel. Similarly, you have

$2\cdot \dfrac{10}{3}+5\cdot \dfrac{4}{3}\neq 0$

So, they aren't orthogonal.

In the second case, we have

$\dfrac{5}{-12}\neq \dfrac{-6}{-10}$

So, they aren't parallel, and

$5(-12)+(-6)(-10)=-60+60=0$

So, they are orthogonal.

Finally, we have

$\dfrac{2}{-4}=\dfrac{-7}{14}=-\dfrac{1}{2}$

So, they are parallel (and thus can't be orthogonal)