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Magnitude of a Scaled Vector

∣kv⃗∣=∣k∣ ∣v⃗∣|k\vec{v}| = |k|\,|\vec{v}|

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

sig figs

Worked example: Scale a length-4 vector by −3 → length 12. 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 Scaled Vector fits

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Unit Vector Component (Normalization)x-Component from Magnitude and Angley-Component from Magnitude and AngleScalar Projection of One Vector onto Another

See also

Cross Product Magnitude: Both use magnitude of the scaled vector.Dot Product from Magnitudes and Included Angle: Both use magnitude of the scaled vector.Magnitude of a 2D Vector: Both use magnitude of the scaled vector.Magnitude of a 3D Vector: Both use magnitude of the scaled vector.Resultant of Two Vectors at an Angle: Both use magnitude of the scaled vector.Scalar Projection of One Vector onto Another: Both use magnitude of the scaled vector.
3 more relatedUnit Vector Component (Normalization): Both use magnitude of the scaled vector.x-Component from Magnitude and Angle: Both use magnitude of the scaled vector.y-Component from Magnitude and Angle: Both use magnitude of the scaled vector.
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Magnitude of a Scaled Vector explained

|v||kv|k

Multiplying a vector by a number scales every component by that number, so the whole arrow stretches or shrinks along its own line and the length scales in step: |kv| = |k||v|. Double a force and you double its magnitude; halve a velocity and the speed halves. The absolute value bars matter. A scalar of −3 applied to a vector of length 4 reverses the direction and gives a length of 12, not −12: magnitudes are never negative, and the minus sign lives entirely in the direction.

Run it backwards and the sign is unrecoverable: a vector of length 5 scaled to length 15 tells you |k| = 3, but k could equally have been −3, since both produce the same length pointing opposite ways. The solver reports the positive value. Scaling by k = 0 collapses the vector to the origin and destroys the information entirely, which is why the original length cannot be recovered from a zero multiplier.

Magnitude of a Scaled Vector formula

∣kv⃗∣=∣k∣ ∣v⃗∣|k\vec{v}| = |k|\,|\vec{v}|
Where
  • ∣kv⃗∣|k\vec{v}|= Magnitude of the scaled vector
  • kk= Scalar multiplier
  • ∣v⃗∣|\vec{v}|= Magnitude of the original vector
Revised 31 October 2026
Cite this page

formula.expert. (2026, October 31). Magnitude of a Scaled Vector Formula & Calculator. https://formula.expert/solvers/scalar-multiplication-magnitude

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