Vector Operations Calculator

A vector represents a quantity with both magnitude and direction. Displacement, velocity, force, electric field, and acceleration are common vector quantities. In coordinates, a vector is written as components along axes, such as x and y in two dimensions or x, y, and z in three.

The magnitude of a vector is its length. In Cartesian coordinates, it follows from the Pythagorean theorem. Components make vectors easy to compute with, while magnitude and direction make them easy to interpret physically or geometrically.

Vector addition combines components. Geometrically, vectors can be placed head-to-tail, and the result runs from the start of the first to the end of the last. Scaling a vector multiplies its magnitude and may reverse direction if the scalar is negative.

These operations model superposition. Multiple forces add to a net force. Multiple displacements add to a final displacement. A velocity can be decomposed into horizontal and vertical parts, then recombined. Component arithmetic is the bookkeeping behind the geometry.

The dot product measures alignment. It equals the product of magnitudes times the cosine of the angle between vectors. A positive dot product means vectors point partly in the same direction, a negative one means partly opposite, and zero means perpendicular. Work in physics uses the dot product because only force along displacement contributes.

The cross product, in three dimensions, produces a vector perpendicular to the two input vectors. Its magnitude equals the area of the parallelogram spanned by them. It appears in torque, angular momentum, orientation, normals, and geometry.

Vector results depend on the coordinate system used for components, but the underlying geometric quantity does not. Rotating axes changes component values while preserving length, angles, and physical meaning.

Good vector work moves comfortably between coordinates and geometry. Coordinates make calculation possible; geometry keeps the result meaningful. When a sign or component seems surprising, drawing the vector often reveals the issue faster than more algebra.