Vector Calculator
Add, subtract and multiply 2D or 3D vectors, and find dot and cross products, magnitudes, unit vectors, the angle between them and projections.
About this tool
Vectors are added and subtracted component by component, and multiplying by a number (a scalar) scales every component. The dot product, A · B = a₁b₁ + a₂b₂ + a₃b₃, is a single number that is 0 when the vectors are perpendicular, and it gives the angle between them through cos θ = A · B ÷ (|A||B|). The cross product A × B, for 3D vectors, is a new vector at right angles to both, whose length is the area of the parallelogram they make.
The magnitude |A| is the vector's length, by Pythagoras, and the unit vector points the same way with length 1. The projection of A onto B is the part of A that lies along B, useful for resolving forces in mechanics. Switch to 2D to see A, B and A + B drawn head to tail; the 2D cross product is shown as a single number, the signed area of the parallelogram.