Show w is orthogonal to line

Question

Show $w$ is orthogonal to the the line $w^Tx + b = 0$

The way we can prove if two vectors are orthogonal is to show that $\vec{a}\cdot\vec{b} = 0$

We can say $w^Tx + b = \begin{matrix} w_1x+b, w_2x+b, ... w_ix+b\end{matrix}$

Since $w^Tx + b = 0$ when we find its dot product with $w$ this should also equal $0$. Is this a valid proof?

Answer 1

In general, the locus of solutions to that equation forms a hyper-plane. To show that $\vec w$ is perpendicular to the hyper-plane we can show that $\vec w\cdot\vec v=0$ for any vector $\vec v$ that lies in the plane.

Any vector in the hyper-plane will have endpoints in the hyper-plane so let $p=(p_1,\dots,p_n)$ and $q=(q_1,\dots,q_n)$ be any two points in the plane. Then $\vec v=(p_1-q_1,\dots,p_n-q_n)$ will be the vector from $p$ to $q$.

$$\vec w\cdot\vec v=\Sigma_{i=1}^{n}w_i(p_i-q_i)=\Sigma_{i=1}^{n}w_ip_i-w_iq_i=\Sigma_{i=1}^{n}-b - -b=0$$

Since this is true for any vector in the plane, $\vec w$ is perpendicular to the plane.