About this tool
Enter two vectors in 2D or 3D and get the full set of vector operations at once: dot product, cross product (in 3D), the magnitude of each vector, the angle between them, the unit vector of the first vector, the projection of one onto the other, and their sum and difference. Nothing is uploaded; the math runs in your browser.
The dot product is the sum of component-wise products, a·b = Σ aᵢbᵢ. In 3D the cross product is (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1). A magnitude is |a| = √(Σ aᵢ²), and the angle between the vectors is acos(a·b / (|a||b|)) converted to degrees. The unit vector is a/|a|, and the projection of a onto b is (a·b / |b|²) b.
Use it for physics, geometry or graphics work. If a vector has zero length the angle, unit vector or projection that depends on it are shown as a dash, since those quantities are undefined.