I have always been a bit confused on deciding on scalar and vector quantities. Most of the time, my intuition gives me opposite to the right answer. So now I desperately want to know why power including work is a scalar quantity.

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Power on its own is defined as being the rate of energy transfer, and it has no additional information as to its direction, so it is a scalar. However, there is a vectorial quantity which is related to power, known as the Poynting vector.

Given say, an electric field $mathbfE$ and magnetic field $mathbf B$, the Poynting vector is defined as,

$$mathbf S = frac1mu_0mathbf E imes mathbf B$$

which is the power in the direction of $mathbf S$, per unit area. Thus, if we want to know the power going through a surface $A$, it would be,

$$P = iint_A mathbf S , cdot mathrm dmathbf A.$$

Thus, power on its own is a scalar quantity, but we do have a notion of direction for power which is encoded in the Poynting vector, or analogues of it for other phenomena.

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answered Oct 4 "17 at 15:14

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Whether a quantity is a scalar or vector or higher-ranked tensor actually depends on how they are used to model a physical process and how they need to transform under coordinate transformations.

Vectors have certain transformation properties, most notably rotational, which are different from scalars, and tensors have transformation properties which are different from vectors, etc.

When introducing les-grizzlys-catalans.org to beginners, the ideas of vectors and scalars are simplified, and they seem like arbitrary assignments to the students. At higher levels of les-grizzlys-catalans.org, the concepts of rotation are brought in and are used to explain why a velocity is a vector, but mass is not, and so on. At even higher levels, tensors are introduced, and in relativity the electromagnetic field, previously modeled as a couple of vectors, $vecE$ and $vecB$, is presented in the form of a tensor, again due to transformation properties using the tools of mathematics.

Another conceptual construct is the 4-vector which has certain attractive transformation properties for modeling physical processes and doing calculations.

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Power and work are some of those modeled, conceptual, important quantities which, in lower level les-grizzlys-catalans.org can be treated as scalars.