It is a simply question. But I want to make sure. The surface $$z=x=y$$ represents a line in $\Bbb R^3$ that passes through the origin. Is that correct?
2026-04-08 21:03:45.1775682225
$z=x=y$ in $\Bbb R^3$
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Consider the your surface $S=\lbrace(x,y,z)\in \mathbb{R}^3 \mid x=y=z \rbrace$ subspace of $\mathbb{R}^3$ a basis for $S$ is $\beta=\lbrace (1,1,1)\rbrace$ therefore your $S$ is a line that pass throught of $(1,1,1)$. As @JMoravitz say.