What is the difference between use of signs('-' and '+') before a vector and a scalar quantity? For example,we can use '-' or '+' sign before Displacement which is a vector quantity and we can also use '-' or '+' sign before Joule which is a scalar quantity. So,what is the difference between the use of signs before vectors and scalars?
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1$\begingroup$ The meaning of the sign depends on what physical quantity your scalar/vector actually denotes. I struggle to see for what answer you're looking in this generality. $\endgroup$– ACuriousMind ♦Mar 10, 2017 at 14:15
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3 Answers
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I don't think this question really has a general answer so I will give a few physics examples from which you can draw conclusions as you wish:
Scalar
- Electric charge or spins in the Ising model: Here the sign affects the type of the "particle/spin/...". A positive charge will be attracted to a negative charge while being repulsed from a positive charge. Similarly spins of like signs will interact differently in the Ising model to spins of opposite signs.
- For quantities like energy,... or others that are essentially defined as a difference between two states, the sign can indicate whether the quantity is consumed or accumulates.
- In some (few) cases a quantity can have positive and negative values without any deeper meaning. This is the case for temperature measured in Celsius or Fahrenheit.
Vector
With vectors, the sign determines the direction. If you change only the sign of a vector, you essentially have it point in the opposite direction. This is true for all kinds of vectors you encounter such as a force changing direction, velocity (moving in opposite direction,...
answered Mar 10, 2017 at 14:50
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Use of the $+$ and $-$ signs before vector and scalar quantities are very simple and elementary. Let me explain them in detail.
For scalars :
Scalars are tensors of rank zero. But in this case, it will be better to consider scalars as just real numbers (complex numbers with a zero imaginary part). In case of scalar quantities, the $+$ and $-$ signs merely indicate the respective positiveness and negativeness of the associated number/magnitude of the scalar.
An example: The temperature at a place can be $5^\circ C$ and again $-5^\circ C$.
However, for scalar quantities like mass, length; negative numbers do not have physical sense. And for other scalar quantities like work, energy (heat energy, potential energy), the $+$ and $-$ signs represent certain conventions.
For vectors :
In case of vector quantities, crudely speaking, we talk about a magnitude as well as a direction. Rather, we can look up a vector as a complex number for this case (and not a first rank tensor). So, if look at a vector in that sense, then you will realize that a vector is just a scalar with an associated direction. The $+$ and $-$ signs before a vector quantity hence point to opposite directions associated with the vector. Here, it must be stated that vectors have an absolute magnitude, by definition.
An example: Let $\vec a=2\hat i$ be a vector in the Cartesian form, with magnitude $2$ and direction $\hat i$. And $-\vec a=-2\hat i$ be a vector in the Cartesian form, with magnitude $2$ and direction $-\hat i$.
answered Mar 10, 2017 at 13:53
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$\begingroup$ Please clear the meaning of these lines in simple language - "Scalars are tensors of rank zero." $\endgroup$ Mar 10, 2017 at 15:05
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$\begingroup$ @user419155 If you are unable to comprehend this part, you can leave it. That doesnt affect the answer much. $\endgroup$ Mar 10, 2017 at 15:55
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The sign on a vector indicates direction, such as left/right, but can be thought of as a simplified expression of an angle (simplified by reducing the angled vector to components). If you can't put an angle on it, it's not a vector (for instance, temperature cannot be at an angle, but it can be negative). The sign on a scalar indicates "more than or less than the amount at reference point." Sticking with temperature, -10 Celsius indicates 10 degrees less molecular motion than the temperature at which water freezes. But notice that there is no negative Kelvin temperature since there cannot be less molecular motion than that reference point.
