Vector Calculator (2D)
Enter two 2D vectors to find their magnitudes, sum, difference, dot product, and the angle between them — with every step shown.
About this calculator
Enter the components of two 2D vectors u = (ux, uy) and v = (vx, vy). The calculator computes each magnitude with the Pythagorean theorem, adds and subtracts the vectors component-wise, finds the dot product u·v = ux·vx + uy·vy, and gives the angle between them via θ = acos(u·v / (|u| |v|)). The angle is undefined when either vector has length 0.
Working with 2D vectors
A 2D vector u = (ux, uy) has magnitude |u| = √(ux² + uy²) — the Pythagorean length of its components. Vectors add and subtract component-wise: u + v = (ux + vx, uy + vy). The dot product u·v = ux·vx + uy·vy measures how much the vectors point the same way; it is zero exactly when they are perpendicular. The angle between two non-zero vectors follows from the dot product: cos θ = (u·v) / (|u| |v|), so θ = acos(u·v / (|u| |v|)). If either vector has length 0 the angle is undefined, because a zero vector has no direction.