Finding all Perpendicular vectors

Question

Given the vector $\vec{a}= (a_1,a_2,a_2)$, what are all of the possible perpendicular vectors?

In the Real number set only.

Answer 1

Given the vector $ \vec{a} $, and arbitrary $\vec{x} = (x,y,z) \in \mathbb{R}^3$, all the possible perpendicular vectors are those $\vec{x}$ that satisfy the equation

$$ \vec{a} \cdot \vec{x} = 0 $$

Or, in other words, all $x,y,z \in \mathbb{R} $ such that

$$ a_1 x + a_2 y + a_3 z = 0 $$