When are vectors $\vec a = (2, 3),\,\vec b = (x, 2)$ perpendicular/parallel

Question

I have the following problem:

Let $\vec a= (2, 3), \vec b = (x, 2)$

When $\vec a$ and $\vec b$ are perpendicular, $x=?$

When $\vec a$ and $\vec b$ are parallel, $x=?$

I have the key with the answers $-3$ and $4/3$ but am wondering which steps I should take to get to that answer? I would also appreciate an explanation of what it means for two vectors to be vertical here because with my current understanding (presumably incorrect) of what it means it could be the case for more than just $x=-3$

Answer 1

Two vectors are vertical (better, they are perpendicular) when their scalar product equals to zero. Thus, $2x + 3 \cdot 2 = 0 \implies x = -3$. The definition of standard scalar product between two vectors is $\langle a, b\rangle = \sum_0^n a_i \cdot b_i$. Two vectors are parallel when one is multiple of the other. Thus, $2 = \tfrac{2}{3}\cdot 3 \implies x = \tfrac{2}{3}\cdot 2 = \tfrac{4}{3}$.

Answer 2

Two vectors are called orthogonal or perpendicular if their scalar product vanishes. Since $\langle \vec{a},\vec{b} \rangle (=\vec{a} \bullet \vec{b})= 2x+6$, orthogonality means $2x+6=0$, i.e. $x=-3$.

The condition for parallelism is that $\lambda \vec{a}=\vec{b}$ for some $\lambda \neq 0$. Then $2\lambda = x$ and $3\lambda=2$. Hence $\lambda = 2/3$ and $x=4/3$.