I have two parallel hyper planes $$a^Tx=b_1,a^Tx=b_2$$ where $a \in \mathbb{R}^n, x \in \mathbb{R}^n ,b \in \mathbb{R}$ and I want to find the distance between the two.
I have read that the distance between the two hyperplanes is also the distance between the two points $x_1$ and $x_2$ where the hyperplane intersects the line through the origin and parallel to the normal vector $\vec a$. These points are given by $$x_1=\frac{b_1}{\|a\|^2_2}a$$ and $$x_2=\frac{b_2}{\|a\|^2_2}a$$
Then the distance is $|x_1-x_2|$ but I don't really understand how we got $x_1$ and $x_2$.
Linear Algebra – Distance Between Two Hyperplanes
linear algebra
Best Answer
Let $x_1$ be any point in the first hyperplane and consider the line $L$ that passes through $x_1$ in the direction of the normal vector $a$. An equation for $L$ is given by $x_1 + at$ for all $t\in\mathbb{R}$. Now find the intersection of $L$ and the second hyperplane:
$$ a^T(x_1 + a t) = b_2 \iff t = (b_2 - a^T x_1)/a^Ta = (b_2 - b_1)/a^T a $$
Therefore the intersection point is $x_2 = x_1 + a(b_2 - b_1)/a^Ta$. The distance between these two points is the distance between the hyperplanes:
$$ \|x_1 - x_2\| = \frac{|b_2 - b_1|}{a^Ta}\|a\| = \frac{|b_2-b_1|}{\|a\|} $$