What is the sign of the magnitude of a vector?

Answer 1

Magnitudes are positive.

When we say that a vector has magnitude and direction, we assume the magnitude is positive and the direction is the actual direction of the vector quantity in question.

This is a convention. If you have a vector of wind velocity that is 20 meters per second in the direction north, then at a later time, the vector is said to be -20 meters per second north, one would know that the actual direction was +20 meters per second south.

That said, the answer to the question is that the magnitude of the vector is taken as positive and the direction is a vector of unit length pointing in the same direction as the given vector. If someone wants to do it differently, that is fine. They would not be wrong.

Answer 2

The - sign convention for the original vectors was created by William Rowan Hamilton as i^2=-1. This was changed by J Willard Gibbs in his Vector Analysis to i^2=+1 in physics.

This + sign convention was created by physicists in the early 1900s to coincide with the physics view that a displacement, di, in the direction of a force fi, is positive energy or work w=fidi=+fd.

This means when a ball falls, then the displacement is -di and the force of gravity is -fi then there is + work w.

The physicists are wrong, work=en-ergy means "input ergs". what is happening when the ball falls is exergy output ergs or exergy and the sign should be w=(-if)(-id)=-fd exergy!

Lifting the ball against gravity is w=-fi(id)=+fd energy ergs in.

There rule adopted by physicists and Gibs makes their positive square rule vectors non-associative. That means for two vectors i and j, (ii)j does not equal i(ij). These positive square vectors are non-assoiciative. Hamilton's original vectors are associative: (ii)j=i(ij)=-j.

Physics + sign convention is the wrong convention mathematically and physically!

Answer 3

The magnitude of a vector is always positive, as it represents the length or size of the vector regardless of its direction.