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I've heard that vector quantities have both magnitude and direction but I've never heard that scalar quantities have magnitude only. Magnitude of vector quantities cannot be negative but what about scalar quantities, like temperature (-1°C)?

If scalar quantities don't have magnitude then what is their "magnitude" called?

Also does the magnitude of a vector quantity include units with the numerical value or only the numerical value?

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A vector quantity, $\vec V,$ can be written as $$\vec V=|\vec V|\ \hat V$$in which $|\vec V|$ is the magnitude of the vector, a scalar quantity which is non-negative. $\hat V$ is the unit vector in the same direction as $\vec V.$

The convention is that $|\vec V|$ is the product of a number and a unit, while $\hat V$ has no unit.

A different sort of scalar arises when we express $\vec V$ as the sum of components, say in the x, y and z directions. Using $\hat i,$ $\hat j$ and $\hat k$ for the unit vectors we can write$$\vec V=V_x \hat i + V_y \hat j+V_z \hat k$$ The scalar coefficients $V_{x},\ V_{y},\ V_z$ can be negative, zero or positive.

"I've never heard that scalar quantities have magnitude only." It is, in fact, quite a common statement in elementary textbooks. Temperature might well be given in such a book as example of a scalar. As you say, (celsius) temperature can be negative, so, clearly, 'magnitude' in this context means real number $\times$ unit, so isn't quite like the magnitude of a vector.

I suspect that temperature wouldn't be given as an example of a scalar in more advanced books, because geometry is not involved in its definition. But this is rather a subtle point.

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  • $\begingroup$ So if we say -1°C what is "-1" here? Is it called something? $\endgroup$ – Sanom Dane Mar 14 at 14:16
  • $\begingroup$ I'd call -1°C the value of the quantity and –1 'the numerical part' of the value'. [In the SI, a quantity (with magnitude only!) is regarded as $\text{number} \times \text{unit}.$] $\endgroup$ – Philip Wood Mar 14 at 14:27
  • $\begingroup$ As another note: we could have defined the vector to have a signed magnitude (which could be negative), and a restricted range for $\hat V$. This can be convenient in scenarios where restricting $\hat V$ simplifies some properties (such as defining $\hat V$ by an angle calculated by $tan^{-1}(\frac{y}{x})$), However, generally speaking using the more typical magnitude and direction has more benefits (magnitude is a more fundamental concept, when you dig into the mathematical side), so it is less common to see this decomposition, especially in dimensions higher than 2. $\endgroup$ – Cort Ammon Mar 14 at 16:49
  • $\begingroup$ In your first equation, how do we rotate the vector 180 degrees? $\endgroup$ – user45664 Mar 14 at 17:07
  • $\begingroup$ By sticking a minus sign in front. On the right hand side we interpret $-\hat V$ as a unit vector in the opposite direction from $\hat V.$ The magnitude, $|\vec V|,$ is unchanged. $\endgroup$ – Philip Wood Mar 14 at 17:11
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A scalar $x$ has magnitude $|x|$, also known as the absolute value. Celarly, $x \neq |x|$ for negative $x$, but instead of saying the scalar has a direction, we would say that it has a sign (+ or -), which is a much simpler concept. Maybe your confusion arises from the fact that a two-dimensional vector can be described through two scalars, one being magnitude and one being direction? Because this means that all magnitudes are scalars, but a negative scalar does not correspond to a magnitude.

As for units: A vector of two-dimensional direction could look like $\bar v = ( 1 \text{ km}, 2 \text{ km})$. Its magnitude is $$|\bar v| = \sqrt{(1 \text{ km})^2 + (2 \text{ km})^2} = \sqrt{5} \text{ km},$$ so yes, the magnitude includes units.

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Intuitive description often makes a thing simple, but the level of understanding could be enhanced by going through a rigorous definition. The importance of rigor over intuitive description is that sometimes the easiest questions becomes trouble some to answer intuitively, like this question.

Intuitive description of the vector may lead to question like :-

What is direction?

What is magnitude?

To understand the matter of what are vector and what are scalar rigorously in mathematics sense, it is advised to go through first few may be 1,2 or 3 chapters in any linear algebra text (like sheldon axler linear algebra, friedberg linear algebra etc), or through the wikipedia article on vector space and fields.

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