Computational Geometry

A point is an ordered pair of coordinates (x, y). A vector also stores (x, y) but represents a displacement (direction + magnitude) rather than a fixed location. The vector from point A to point B is B − A = (Bx − Ax, By − Ay).

Add componentwise: (ax, ay) + (bx, by) = (ax + bx, ay + by). Geometrically this is the tip-to-tail rule (the diagonal of the parallelogram spanned by a and b).

s·(x, y) = (s·x, s·y). It stretches the vector's length by |s| and keeps the same direction when s > 0, reverses direction when s < 0. Scaling by 0 gives the zero vector.

|v| = √(x² + y²) by the Pythagorean theorem. To avoid the square root and floating point, compare squared lengths x² + y² when only ordering matters.

a · b = ax·bx + ay·by. Equivalently a · b = |a||b|cos θ, where θ is the angle between them. It is a scalar.

Because a · b = |a||b|cos θ: positive means the angle is acute (< 90°), zero means perpendicular (= 90°), negative means obtuse (> 90°).