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Magnitude of a Scaled Vector

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

Worked example: Scale a length-4 vector by −3 → length 12 — press Try an example to run it live, then adjust anything.

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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

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