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Magnitude of a 2D Vector

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

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

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Worked example: Components (3, 4) → magnitude 5. Press Try an example to run it live, then adjust anything.

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Unit Vector Component (Normalization)x-Component from Magnitude and Angley-Component from Magnitude and AngleMagnitude of a Scaled Vector
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See also

Magnitude of a 3D 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.Dot Product of Two 3D 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.
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Magnitude of a 2D Vector explained

|v|vyvx

A vector is an arrow: a direction plus a length. Drop it onto axes and it splits into two ordinary numbers, the components vx and vy, which form the legs of a right triangle whose hypotenuse is the arrow itself, so the magnitude is √(vx² + vy²). A hiker who walks 3 km east and 4 km north ends up 5 km from camp, not 7 km, because displacement is a vector and its parts add head-to-tail rather than arithmetically. That gap between 5 and 7 is the single most common student error in the whole subject.

Run it backwards and a magnitude plus one component recovers the other: a vector of length 13 with vy = 5 must have vx = √(169 − 25) = 12. The solver returns the positive root, since a component's sign is a matter of which way the axis points and the arithmetic cannot know it. Note also that no component may exceed the magnitude: a leg longer than the hypotenuse describes a triangle that does not exist.

Magnitude of a 2D Vector formula

∣v⃗∣=vx2+vy2|\vec{v}| = \sqrt{v_x^2 + v_y^2}
Where
  • ∣v⃗∣|\vec{v}|= Vector magnitude
  • vxv_x= x-component
  • vyv_y= y-component
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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
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formula.expert. (2026, October 31). Magnitude of a 2D Vector Formula & Calculator. https://formula.expert/solvers/vector-magnitude-2d

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