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 $\mathbf{E}$ and magnetic field $\mathbf B$, the Poynting vector is defined as,

$$\mathbf S = \frac{1}{\mu_0}\mathbf E \times \mathbf B$$

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,

$$P = \iint_A \mathbf S \, \cdot \mathrm d\mathbf A.$$

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

JamalSJamalS
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When introducing moment-g.com to beginners, the ideas of vectors and scalars are simplified, and they seem like arbitrary assignments to the students. At higher levels of moment-g.com, 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, $\vec{E}$ and $\vec{B}$, is presented in the form of a tensor, again due to transformation properties using the tools of mathematics.