Vector Operations Explained

Vectors describe quantities that have both size and direction. In physics and engineering they represent displacement, velocity, force, and momentum; in machine learning they represent data points in feature space. This calculator performs the four fundamental 3D vector operations in one pass.

The Four Operations

Addition and subtraction are component-wise: A + B = (ax+bx, ay+by, az+bz). The dot product A·B = ax·bx + ay·by + az·bz is a scalar measuring directional agreement, and the cross product produces a perpendicular vector whose magnitude equals the parallelogram area |A||B|sin(θ).

From Dot Product to Angle

Rearranging the dot-product identity gives cos(θ) = A·B ÷ (|A||B|), the standard way to find the angle between two vectors in 3D. When the dot product is zero the vectors are perpendicular; when it is positive they point mostly the same way, and when negative they point mostly apart.