I know that the equation of a hyperplane in n-dimensions is given by: $$w^Tx+w_0=0$$ Where $w$ is a vector that is perpendicular to the surface of the hyperplane and $w_0$ is a constant. I also know that if $w_0=0$, then the plane passes through the origin. My question is, what is the physical significance of the constant term $w_0$. Is it the distance of the plane to the origin? If not then what is its physical significance?
2026-04-09 16:57:27.1775753847
Equation of a hyperplane in n- dimensions.
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Assume that the length of $w$ is $1$.
Since $w$ is perpendicular to the hyperplane, the orthogonal projection of the origin to the hyperplane is $\lambda w$ for a unique real number $\lambda$.
The length of $\lambda w$ is $|\lambda|$, this is the distance of the origin to the hyperplane, and since $\lambda w$ is assumed to be on the hyperplane, it satisfies $$w^T(\lambda w)+w_0=0\implies \lambda=-w_0$$ since $w^Tw=\|w\|^2=1$ by assumption.