Vector Calculator

The dot product is Σ aᵢbᵢ, and it is the one that carries the geometry: cos θ = a·b ÷ (|a||b|) gives the angle, its sign alone tells you whether that angle is acute, obtuse or right, and dividing by |b| gives the length of a along b. From there the projection of a onto b and the rejection — what is left, perpendicular to b by construction — split any vector into a component along a direction and a component across it, which is most of what vectors get used for. It works in any number of dimensions, so the page does not restrict it to two or three.

The cross product is different: it is only defined in three dimensions (and seven, which nobody asks for). Its magnitude is the area of the parallelogram the two vectors span, its direction is perpendicular to both by the right-hand rule, and reversing the order negates it. Asking for one in 2-D is refused rather than quietly padded with a zero — the usual 2-D substitute, a₁b₂ − a₂b₁, is a scalar, not a vector, and the difference matters. The scalar triple product a·(b × c) follows from both: it is the signed volume of the parallelepiped, zero when the three vectors are coplanar.

Plain Euclidean vectors in double precision, components taken as given in an orthonormal basis. No units are attached, so nothing here knows whether you are adding metres or newtons; no coordinate transform, no basis change and no curvilinear system. Normalising a very short vector loses precision, because the magnitude is a square root of a sum of small squares. Whether two vectors count as parallel or perpendicular is decided against a tolerance you set and can see, not a hidden one. Nothing leaves the browser; the same four anonymous usage counts as the rest of the site apply.