Interactive lessons

Vector Calculus
from first principles

Each lesson is an interactive tool you can fiddle with — drag sliders, watch the geometry move, build intuition before touching a formula. Start at the beginning or jump to any topic.

Foundations

— 6 of 7 complete
(x, y) Point Segment Vector
Foundations — 01
From Point to Vector
Four stages, two draggable points. Watch a coordinate become a line segment, discover direction, then gain magnitude to become a true vector.
interactive · 2D · draggable →
x y z sqrt(a2+b2) c ||v||
Foundations — 02
Vector Magnitude
From a 1D number line to full 3D Euclidean distance. Four stages building up to ||v|| = sqrt(a² + b² + c²).
interactive · 1D → 3D →
u v u+v
Foundations — 03
Vector Addition
Tip-to-tail with animation, the parallelogram rule, and component arithmetic — four stages showing addition three different ways.
interactive · 2D · animated →
u shadow v u·v
Foundations — 04
The Dot Product  (u · v)
From the angle between two vectors to the shadow, to the formula. Four stages building intuition without assuming any trigonometry.
interactive · 2D · draggable →
u v u×v
Foundations — 05
The Cross Product  (u × v)
From parallelogram area to a perpendicular vector rising out of the plane. Includes the right-hand rule and the degenerate zero case.
interactive · 2D + 3D · draggable →
dy/dx = 2 y = x² + 2 2D curve 3D surface Vec. field
Calculus — 06
Partial Derivatives & Gradients
From a 2D slope arrow on a curve, to a 3D orbitable surface, to a full gradient vector field — four stages building up to ∇f.
interactive · 2D + 3D · Three.js →