How this calculator works
Vectors combine length and direction. This workbench makes the operation explicit: addition and subtraction return component vectors; the dot product returns a scalar; the cross product returns a perpendicular 3D vector. Normalisation retains a direction with length one, and projection retains the part of A parallel to B. Point mode returns the directed displacement B-A. Planar polar inputs are converted to components first. Component values must share a common Cartesian basis and consistent units.
Formula
A+B=(Ai+Bi); displacement=B-A; A dot B=sum(Ai Bi); unit vector=A/|A|; projection of A onto B=(A dot B/|B|^2)B; angle=acos((A dot B)/(|A||B|)).
Worked example
For A=(2,2) and B=(3,0), the projection of A onto B is (2,0), while their dot product is the scalar 6. Normalising A gives approximately (0.707106781187,0.707106781187). The angle between the vectors is 45 degrees.
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Frequently asked questions
Why do some operations return one number?
The dot product is a scalar, and an angle is an angular measurement. Addition, cross product, normalisation and projection return vectors.
Do positions and vectors mean the same thing?
A position identifies a location relative to an origin. Subtracting two positions gives the displacement vector between them.
Can the zero vector be used?
It is valid for addition, subtraction, dot product and cross product. It has no direction for normalisation or angle calculations, and cannot be the target of projection.
When should I use the specialised tools?
Related calculators provide focused tasks such as scaled combinations, inverse components, cross-product recovery and detailed dot-product projections.