Vector Magnitude Calculator

Vector magnitude is the length of a vector, independent of the direction in which it points. It is also called the Euclidean norm or simply the norm. For a vector with components x, y, and z, magnitude is the square root of x squared plus y squared plus z squared. The formula is an extension of the Pythagorean theorem from a right triangle into three-dimensional coordinate space. For a two-dimensional vector, set z to zero. A familiar example is the vector (3, 4), whose magnitude is 5 because 3 squared plus 4 squared equals 25. In three dimensions, the same logic first combines perpendicular distances in the xy-plane and then incorporates depth along the z-axis. The vector (2, 3, 6), for example, has magnitude 7 because its squared components sum to 49. Magnitude is always nonnegative. Changing every component's sign reverses the vector's direction but leaves its length unchanged, because squaring removes the signs. The zero vector is the only vector with magnitude zero. Decimal, fractional, and negative components are all valid inputs, provided every coordinate uses the same unit and coordinate scale. This measurement appears throughout mathematics, physics, engineering, data science, and computer graphics. A velocity vector's magnitude is speed. A force vector's magnitude is the force strength. A displacement vector's magnitude is straight-line distance between its start and end points. In machine learning and numerical analysis, norms measure the size of parameter vectors, errors, and distances between observations. Magnitude is also used to normalize a vector. Dividing each component of a nonzero vector by its magnitude produces a unit vector that points in the same direction but has length one. Unit vectors make it easier to separate direction from scale. Magnitudes also appear in dot-product angle formulas and in tests of orthogonality and convergence. The calculator uses the built-in hypotenuse operation for stable floating-point evaluation and displays up to six decimal places. Exact square-root forms may be preferable in symbolic algebra, while the decimal result is convenient for measurement and applied work. Use zero for a missing dimension rather than leaving the field blank so the calculator can distinguish a complete 2D vector from incomplete input.