The velocity vector can change in magnitude and in direction or both at once. Hence, the acceleration accounts for both the rate of change of the magnitude of the velocity vector and the rate of change of direction of that vector. The same reasoning used with respect to the position of a particle to define velocity, can be applied to the velocity to define acceleration. The acceleration of a particle is the vector defined by the rate of change of the velocity vector. The average acceleration of a particle over a time interval is defined as the ratio. a ¯ ¯ = Δ Δ v ¯ ¯ Δ Δ t = Δ Δ v ¯ ¯ x Δ Δ t x ^ ^ + Δ Δ v ¯ ¯ y Δ Δ t y ^ ^ + Δ Δ v ¯ ¯ z Δ Δ t z ^ ^ = a ¯ ¯ x x ^ ^ + a ¯ ¯ y y ^ ^ + a ¯ ¯ z z ^ ^ where Δ v is the average velocity and Δ t is the time interval.