Computational Geometry

How is a point in the plane represented, and how does it relate to a vector?

A point is an ordered pair of coordinates (x, y). A vector also stores (x, y) but represents a displacement (direction +

How do you add two vectors, and what does the sum represent geometrically?

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

What does scaling a vector by a scalar s do?

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

How is the magnitude (length) of a vector computed?

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

What is the dot product of two vectors, and how is it computed?

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