Can somebody explain why the orthogonal complement of a plane is a line? Why is it not another plane perpendicular to the first one?
2026-03-28 22:37:07.1774737427
Orthogonal complement of a plane
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The orthogonal complement satisfies $U\cap U^\perp=\{\vec 0\}$. Two planes, assuming distinct, which pass through the origin will intersect in a line which passes through the origin. So the orthogonal complement cannot be a plane. So it must be either a line or just $\{\vec0\}$. But the normal vector to your plane of course belongs to the orthogonal complement, so it cannot be $\{\vec 0\}$.