Vector Calculator

A vector represents a quantity with magnitude and direction. In Cartesian coordinates, a three-dimensional vector is written as an ordered triple of x, y, and z components. This calculator performs four common operations on two vectors: addition, subtraction, the dot product, and the cross product. Entering zero for both z-components also lets you handle ordinary two-dimensional addition, subtraction, and dot products. Addition and subtraction work component by component. To add vectors A and B, add their x-coordinates, then their y-coordinates, and finally their z-coordinates. Subtraction follows the same pattern but subtracts each component of B from A. These operations model combined or relative displacement, force, velocity, acceleration, and many other physical quantities. The dot product multiplies corresponding components and adds the products. Its result is a scalar rather than a vector. For A = (Ax, Ay, Az) and B = (Bx, By, Bz), the result is Ax times Bx plus Ay times By plus Az times Bz. A zero dot product means nonzero vectors are perpendicular. The dot product also connects vectors to angles because it equals the product of their magnitudes and the cosine of the angle between them. The cross product is defined for three-dimensional vectors and returns another vector. Its result is perpendicular to both inputs, with direction determined by the right-hand rule. The magnitude of the cross product equals the area of the parallelogram spanned by the two vectors. Cross products are used in torque, angular momentum, surface normals, computer graphics, and geometry. Reversing the input order reverses the sign of the result. Use consistent units and coordinate axes for meaningful results. Addition and subtraction require quantities of the same type, while dot and cross products produce derived units. The calculator uses standard floating-point arithmetic and displays concise decimal results. It provides the computed value directly so you can verify homework, explore examples, or continue a longer linear algebra calculation without repeating component arithmetic by hand.