Vector Addition Calculator

Vector addition combines quantities that have both magnitude and direction. Instead of adding entire vectors as single numbers, you add their corresponding coordinates. For vectors A = (Ax, Ay, Az) and B = (Bx, By, Bz), the sum is A + B = (Ax + Bx, Ay + By, Az + Bz). This component-by-component rule works because each coordinate describes an independent direction in the chosen coordinate system. In two dimensions, a vector has horizontal and vertical components. You can still use this calculator by entering zero in both z-component fields. In three dimensions, the z-coordinate adds depth, so all three pairs of components contribute to the resultant. Negative components are valid and indicate direction along the negative side of an axis. Decimal values are also supported for measurements that are not whole numbers. Vector addition has a clear geometric meaning. Place the tail of the second vector at the head of the first vector. The resultant runs from the tail of the first vector to the head of the shifted second vector. This is called the head-to-tail method. For two vectors drawn from the same starting point, the same result appears as the diagonal of the parallelogram formed by the vectors. The operation is commutative, meaning A + B equals B + A, and associative, meaning vectors can be grouped in any order when adding three or more of them. Those properties make it practical to combine forces, velocities, displacements, electrical fields, and other directed quantities. However, coordinates must refer to the same axes and use compatible units. A displacement measured in meters cannot be added directly to one measured in feet without conversion. This calculator reports the exact component sum using ordinary arithmetic. It does not separately calculate the resultant magnitude or direction, although those can be found from the resulting coordinates. In three dimensions, magnitude is the square root of x squared plus y squared plus z squared. Use the displayed sum as the starting point for that calculation or for later dot products, cross products, projections, and coordinate transformations.