This is technically about a homework problem, but I'm only asking for clarification on the statements made, not about solutions. Keep in mind I've had little practical experience with linear algebra.
I'm given a statement about the distance from a point to a hyperplane as well as an image to illustrate it:
In binary classification, the distance $d$ of a point $\mathbf{x}$ to a hyperplane $\mathbf{w}$ is computed by the length of the projection of $\mathbf{x}$ onto $\mathbf{w}$, minus the distance $r$ to the origin: $$ d = \mathbf{x} \cdot \frac{\mathbf{w}}{\| \mathbf{w} \| } - r $$
I'm fine with the equation, but I'm not sure how a single vector w is used to uniquely define a hyperplane.
I was assuming w is interpreted two ways: 1) as pointing from the origin to a point on the plane, and 2) as a normal vector for that plane. However, if that were true, shouldn't w should be exactly as long as $r$? In the image it's definitely not.
The text goes on to state that the length of w doesn't matter, but if w were used to define the location of the hyperplane it definitely would.
In conclusion, what's wrong with my assumption of how one vector defines a hyperplane, and how am I supposed to interpret it instead?
$w$ is the normal direction of the hyperplane.
The length of $w$ doesn't matter because in the formula, we are going to normalize it anyway, that is $\frac{w}{\|w\|}$.
A hyperplane is defined in the form of $w^Tx+d=0$ and the distance can be calculated by $r = \frac{|d|}{\|w\|}$.