newtonian mechanics - Can magnitude be negative?

My teacher told that magnitude is the positive value of that quantity or the modulus of that quantity.

he also told that vector quantities have both magnitude and direction and scalar quantities have only magnitudes and hence are always positive.

However, gravitational potential energy is always negative except for being 0(at infinity)

But gravitational potential energy is also a scalar quantity.

So is there magnitude negative?

What I thought about it was that it's magnitude is negative.

Let's take an example of any vector quantity,say velocity.

If a body is moving with the velocity of -5m/s, that means it is moving with a speed of 5m/s in the direction opposite to the positive direction. And here, the body is covering 5 meters every second, though it's velocity is -5m/s.

But if a body has potential energy -40J, it does not mean that it has actual potential energy 40J but is in opposite direction, hence the magnitude should be negative

Please tell me that will the magnitude be positive or negative?

Answer

This is a very common misconception among physics students, so let me see if I can provide some examples that will make the distinction clearer.

VECTORS are quantities that have a magnitude and a direction. The magnitude of the velocity is speed, which is always positive.

• Examples: As you pointed out, one of the simplest examples of a vector quantity is velocity. Other good examples are forces, and momenta.


• For a vector $\vec{v}$, the magnitude of the vector, $|\vec{v}|$ is the length of the vector. This quantity is always positive! The magnitude of velocity, for example, is speed, which is always positive. (If a car is traveling 95 mph, A radar gun would register the speed of a car as 95 mph regardless of whether the car was going backwards, forwards, or sideways). Similarly, the magnitude of a force is always a positive number, even if the force points down. If you have $7$ N forces point up, down, left and right, the magnitude of those forces are all just $7$ N. Once again, the magnitude of a vector is its length, which is always positive.

SCALARS on the other hand work entirely differently. Scalar quantities have a numerical value and a sign.

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  Examples: Temperature is a nice simple example. Others include time, energy, age, and height.

  
  


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  For a scalar $s$, the absolute value of the scalar, $|s|$ is simply the same numerical value as before, with the negative sign (if it existed) chopped off. We do not (or at least we shouldn't!) talk about the "magnitude" of a scalar! Conceptually, I recommend thinking about the absolute value of a scalar, and the magnitude of a vector as completely different things. If it is $-3°F$ outside, it does not make sense to talk about the magnitude of the temperature. You could, however, compute the absolute value of the temperature to be $3°F$.

  
  


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  Note that some scalar quantities don't make sense as negative numbers: A person's age is a scalar quantity, and we don't really talk about negative age. Another example is temperatures measured on the kelvin scale.

  
  

So, to answer your question, energy is a scalar, so it does not have a magnitude. If a body has -40J of potential energy, then it simply has 40J less than your arbitrary 0 point. It does not make sense to talk about the magnitude of this scalar quantity. Please let me know if that helped or hurt your understanding!