Suppose I have an $n$ dimensional hyperplane defined by the equation
$\{\mathbf x\mid \mathbf a^T \mathbf x = b\}, \mathbf a \in \mathbb R^n, b \in \mathbb R$
How do I find the equation of the normal to this hyperplane and its intersection to the hyperplane?
The normal line (that is, the orthogonal subspace of the hyperplane) is the span of the vector $a$. To find the intersection solve $$y=\lambda a,$$ $$<a,y>=b$$ which gives $\lambda=\frac{b}{|a|^2}$ and $y=\frac{ba}{|a|^2}$.