$X = \{x_1, x_2, x_3: x_1 + x_2 + x_3 = 0\}$
It implies,
$X = \{-x_2 - x_3, x_2, x_3: x_2, x_3 \in R^3\}$
which can also be expressed as,
$\{x_2(-1, 1, 0) + x_3(-1, 0, 1): x_2, x_3 \in R^3\}$.
Equivalently, it also holds as,
$\{x_2(1, -1, 0) + x_3(1, 0, -1): x_2, x_3 \in R^3\}$.