Coordinate Geometry

The Coordinate Plane

The Cartesian coordinate system, invented by René Descartes, assigns every point in the plane a unique pair (x, y). This seemingly simple idea is one of the most important in all of mathematics — it lets us use algebraic equations to describe geometric shapes.

Distance & Midpoint

The distance formula is a direct consequence of the Pythagorean theorem. It generalizes to n dimensions in linear algebra: d = ‖v₁ − v₂‖.

Lines & Slope

Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals: m₁ · m₂ = −1.

The concept of slope is the geometric precursor to the derivative. In calculus, we ask: what is the slope of a curved line at a single point?

Conic Sections

The four curves obtained by cutting a cone with a plane — each has a standard equation on the coordinate plane:

Parabolas arise in quadratic equations and physics (projectile motion). Ellipses describe planetary orbits (Kepler's first law). Hyperbolas appear in navigation systems and special relativity.

Transformations

Geometric transformations can be expressed algebraically using coordinates:

• Translation by (a, b): (x, y) → (x + a, y + b)
• Reflection over x-axis: (x, y) → (x, −y)
• Reflection over y-axis: (x, y) → (−x, y)
• Rotation by θ: (x, y) → (x cos θ − y sin θ, x sin θ + y cos θ) — uses trigonometry
• Dilation by factor k: (x, y) → (kx, ky)

The rotation formula uses sine and cosine. In linear algebra, all these transformations are represented as matrix multiplication — an incredibly powerful unification.