Power is the capacity of doing work and it can be expressed in the following way: Energy = Power x Time, or Power = Energy/Time.

For the sake of the above expressions, let us assume that energy is kinetic or potential.

Kinetic energy = 1/2 mv^2; where mass (M) is constant and velocity (V) always has direction. This actually gives us a value of energy/time as a vector quality.

Potential energy such as MGH, for example, acceleration due to gravity, has a direction (downwards!) and is, therefore, a vector quality whilst still retaining scalar qualities.

As a further example, power can be expressed in the following way:

E x I = I^2*R Watts has a vector quantity due to it possessing a direction.

It is also possible to consider work as a scalar value. With constant force exerted through constant direction, it is equivalent to a dot product, alternatively referred to as the scalar or inner product.

Inconstant force or one applied along changing directions, means that work becomes known as the dot product's integral, and is expressed as: (F(r) * dr).

This again gives a scalar result, with a vector that has been displaced represented by dr.

A further example is a ball thrown forward. It travels in arc motions as it moves forward, and down as gravity takes hold of it. Gravity only affects the movement of the object as it goes down: This is the only part that counts. The dot product covers this, ignoring all other vector components or other routes.