formula.expertYour record

Magnitude of a 3D Vector

∣v⃗∣=vx2+vy2+vz2|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}

Units aren’t used in this calculation: every value is a plain number.

sig figs

Worked example: Components (2, 3, 6) → magnitude 7. Press Try an example to run it live, then adjust anything.

Enter your known values, leave one input blank, and solves for the missing one. Tap a variable’s symbol to see what it means, with a typical value.

Where Magnitude of a 3D Vector fits

Missing an input?
Your answer feeds
Unit Vector Component (Normalization)x-Component from Magnitude and Angley-Component from Magnitude and AngleMagnitude of a Scaled Vector
In this toolkit

See also

Dot Product of Two 3D Vectors (Components): Both use vector x component and vector y component.Magnitude of a 2D Vector: Both use vector magnitude and vector x component.Angle Between Two 2D Vectors (Components): Both use vector x component and vector y component.Cross Product z-Component of Two 2D Vectors: Both use vector x component and vector y component.Direction Angle of a 2D Vector: Both use vector x component and vector y component.Dot Product of Two 2D Vectors (Components): Both use vector x component and vector y component.
8 more relatedy-Component from Magnitude and Angle: Both use vector magnitude and vector y component.Unit Vector Component (Normalization): Both use vector magnitude and vector x component.x-Component from Magnitude and Angle: Both use vector magnitude and vector x component.Cross Product Magnitude: Both use vector magnitude.Dot Product from Magnitudes and Included Angle: Both use vector magnitude.Magnitude of a Scaled Vector: Both use vector magnitude.Resultant of Two Vectors at an Angle: Both use vector magnitude.Scalar Projection of One Vector onto Another: Both use vector magnitude.
Learning zone

Magnitude of a 3D Vector explained

vxvzvy|v|

In space the same rule simply picks up a third term: apply Pythagoras once in the xy-plane, then again with that diagonal and the z-leg, and the two steps collapse into √(vx² + vy² + vz²). It is the length of the space diagonal of a box with sides vx, vy, vz. Every phone in a pocket computes it thousands of times a second: a three-axis accelerometer at rest reports something like (0.20, −1.10, 9.72) m/s², and the magnitude √(0.04 + 1.21 + 94.48) ≈ 9.79 m/s² is gravity, whatever way the handset happens to be tilted.

The components (2, 3, 6) are a favourite of textbook writers because 4 + 9 + 36 = 49 lands exactly on 7. Solving backwards for a missing component takes the positive root and requires the other two to leave something under the radical: if vx² + vy² already exceeds |v|², the vector you described cannot be built.

Magnitude of a 3D Vector formula

∣v⃗∣=vx2+vy2+vz2|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}
Where
  • ∣v⃗∣|\vec{v}|= Vector magnitude
  • vxv_x= x-component
  • vyv_y= y-component
  • vzv_z= z-component
In this toolkit

Vector operations

12 formulasMagnitude, direction, components and unit vectors, plus the dot and cross products and the projections and angles they give you.

Revised 31 October 2026
Cite this page

formula.expert. (2026, October 31). Magnitude of a 3D Vector Formula & Calculator. https://formula.expert/solvers/vector-magnitude-3d

formula.experthttps://formula.expert/solvers/vector-magnitude-3d