With two features, a linear classifier splits the plane into two regions with a single line — the decision boundary. It is exactly the set of points where the model is perfectly undecided, scoring 0.5:

On one side \vec{w}\cdot\vec{x} + b > 0 and the model predicts class 1; on the other it's negative and predicts class 0. The weight vector \vec{w} points perpendicular to the boundary, toward the positive class — a direct echo of the dot product's sign.

Steer the boundary with the weights

The two weights and the bias are the boundary. Adjust them and watch the dividing line swing and shift, the shaded half-planes following along. The arrow is the weight vector \vec{w}, always at right angles to the boundary and pointing into the class-1 region.

Straight lines and beyond

A linear boundary is a straight line (a flat hyperplane in higher dimensions) — simple, fast, and often enough. When classes interlock in ways no line can separate, you reach for curved boundaries: add polynomial features, or let a neural network bend the boundary into whatever shape the data demands. But every one of them is still "which side are you on?" — just with a fancier dividing surface.