Grab an arrowhead and drag

➡️ Vector Playground

Drag two arrows on the grid and every number updates live: sum and difference by the parallelogram rule, dot product with the angle between them, 2D cross product, lengths and the projection of a onto b — the picture and the formulas always agree.

a=(,) b=(,)

Dot product a·b

Angle θcosθ = a·b/(|a||b|)

Cross a×b (z)

|a| · |b|√(x²+y²)

a + b

proj of a on b(a·b/|b|²)·b

Dot product > 0 → the vectors lean the same way; = 0 → perpendicular; < 0 → opposing. The 2D cross product a×b = ax·by − ay·bx is the signed area of the parallelogram — positive when b lies counter-clockwise from a.