Why is kinetic energy not a vector, though it uses velocity in its definition of : K = 1/2mv^2
Imagine 4 particles. One traveling 10m/s to the north, one 10 m/s to the south, one 10 m/s to the east, and one 10 m/s to the west. Is there any reason they should have different energies from each other? No of course not. There is no special preference given to any spatial direction. So kinetic energy does not depend on direction, hence it must be a scalar, not a vector. There is nothing that prohibits defining scalars in terms of vectors, although incidentally, the v in kinetic energy is speed (|v|, a scalar), not velocity.
KE is a magnitude. Momentum would be a vector. Why is the distance between two points not a vector? Same reason KE isn't.
Actually, it is defined as the work required to make a stationary particle of mass m to move at velocity v, where v is velocity. See here for more information.
Your confused with the difference between velocity and speed. Kinetic energy does not use velocity in the equation you stated. That equation uses speed, the magnitude of a velocity vector. Since the magnitude of a vector is a scalar, there is no vector term in the kinetic energy equation.
Also, as Hootenanny pointed out, that is not the definition of kinetic energy. See his post for more info.
The change in kinetic energy is equal to the work done by a conservative force. So KE and work must have the same units and type. Work is the dot product of force and displacement. Force and displacement are vectors, and the dot product of two vectors is a scalar. Therefore KE must also be a scalar. In fact, another way to write the expression for KE is 1/2 m v.v which makes it clear that it is a scalar.
The square of a vector quantity is almost always actually the dot product of the vector with itself, a scalar quantity.
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