Can magnitude be negative?

Scalar quantities can be negative. Instead of saying "scalar quantities have only magnitudes," a better description might be that a scalar quantity can be described using only one number per point in space. That number may be positive or negative.

In contrast, a vector quantity cannot be described using only one number per point in space. In 3-d space, we need 3 numbers per point in space to describe a vector quantity.

The word "magnitude," whether applied to a scalar or a vector or anything else, normally refers to a non-negative number. It is sometimes used to refer to the absolute value of a scalar, and sometimes used to refer to the norm (e.g., length) of a vector.

In summary, this is how the words are typically used:

• Scalar typically refers to a single element of a number field (or a single element per point in space), such as a real number (which can be positive or negative) or even a complex number (this is common in the context of quantum physics).

• Magnitude typically refers to a non-negative real number.

The real culprit here is statements like "a vector has both magnitude and direction, but a scalar has only magnitude." The last part of that statement either (1) imposes an unconventional restriction on the usage of the word "scalar," or (2) exercises unconventional freedom in the usage of the word "magnitude."

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.

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

• 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 $3°F$.

• 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!