Magnitude of a Scaled Vector

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.