How can you prove mathematics that power is a scalar quantity
Power is actually not a scalar quantity, but a physical quantity known as a vector quantity. A scalar quantity is a physical quantity that only has magnitude, whereas a vector quantity has both magnitude and direction.
Power is defined as the rate at which work is done or energy is transferred. Mathematically, power (P) is calculated as the product of force (F) and velocity (v), or equivalently the product of force and the rate of change of displacement (s) with respect to time (t):
P = F·v = F·(ds/dt)
Both force and displacement are vector quantities, giving power a vector nature.
Furthermore, power has units of joules per second (J/s) or watts (W), which also suggest its vector nature. Scalars, on the other hand, are expressed in units without a direction.
Therefore, it can be concluded that power is not a scalar quantity but a vector quantity based on its mathematical definition, the presence of vector quantities in its equation, and its units.