The difference of two vectors in a hyperplane also lies in the hyperplane?

Question

Let $H = \{x | a^Tx + c = 0\}$ be a hyperplane as per definition in Wolframalpha https://mathworld.wolfram.com/Hyperplane.html

Then is it true that if $x1 \in H$, $x2 \in H$, then $x1-x2 \in H$?

I think it is true but I am missing something.

$x1 - x2 \in H$ if $w^T (x1 - x2) + c = 0$.

$w^T(x1 - x2) + c = w^Tx1 - w^Tx2 + c = -c + -w^Tx_2 + c = c \neq 0$.

Am I missing something?

Answer 1

Clearly not, unless the hyperplane passes through the origin, as you’ve already computed. The difference vector is parallel to the hyperplane, though. Think about a line in $\mathbb R^2$: does the difference of two vectors on the line lie on it?