3D Vector Magnitude Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/10/2026
The magnitude of a 3D vector with components (x, y, z) is calculated with ‖v‖ = √(x²+y²+z²), the direct generalization of the Pythagorean theorem to three dimensions. For the vector (2, 3, 6), the magnitude is exactly 7.
Explanation
A vector's magnitude measures its geometric length, independent of its direction. In three dimensions, it is obtained by applying the Pythagorean theorem twice: first to find the length of the vector's projection in the xy plane (√(x²+y²)), then again by combining that intermediate length with the z component to get the real length in space — which, once the two steps are combined, gives the direct formula √(x²+y²+z²). This quantity appears in many vector calculations: it serves for example as the denominator in calculating the angle between two vectors from the dot product (θ = arccos(u·v ÷ (‖u‖×‖v‖))), and it directly measures the distance between two points in space if you take the difference of their coordinates as the vector components. In physics, the magnitude of a velocity or force vector gives the scalar value of that quantity (speed "in km/h", force "in newtons") independent of its direction in space.
Example: magnitude of the vector (2, 3, 6)
Inputs
Vector v = (2, 3, 6).
Calculation
‖v‖ = √(2² + 3² + 6²) = √(4 + 9 + 36) = √49 = 7.
Result
The magnitude of this vector is exactly 7.
Frequently asked questions
Can the magnitude be negative?
No, never: the magnitude is a square root of a sum of squares, so always positive or zero, whatever the values (positive or negative) of the vector's x, y, and z components. Only the zero vector (0,0,0) has a magnitude exactly equal to 0.
What is the difference between a vector's norm and its length?
None: they are two names for the same quantity. "Norm" is the preferred term in linear algebra (it generalizes to spaces where the geometric intuition of "length" is less direct), while "length" or "magnitude" are more common in physics and everyday use, but the formula and result are identical.
How do you calculate a vector's magnitude in only two dimensions?
The same formula applies by simply dropping the z component: ‖v‖ = √(x²+y²), the Pythagorean theorem applied once in the plane. It is a special case of the 3D formula when z = 0.