The velocity of a particle is a vector quantity that describes the direction as well as the magnitude of motion of the particle. More mathematically, the rate of change of the position vector of a point with respect to time is the velocity of the point. Consider the ratio formed by dividing the difference of two positions of a particle (displacement) by the time interval. This ratio is called the average velocity over that time interval and is defined as v ¯ ¯ = Δ Δ r Δ Δ t = Δ Δ x Δ Δ t x ^ ^ + Δ Δ y Δ Δ t y ^ ^ + Δ Δ z Δ Δ t z ^ ^ = v ¯ ¯ x x ^ ^ + v ¯ ¯ y y ^ ^ + v ¯ ¯ z z ^ ^ where Δ Δ r is the displacement vector during the time interval Δ Δ t . In the limit that the time interval Δ Δ t approaches zero, the average velocity approaches the instantaneous velocity, defined as the time derivative of the position vector, v = lim Δ Δ t → → 0 Δ Δ r Δ Δ t = d r d t = v x x ^ ^ + v y y ^ ^ + v z z ^ ^ . Thus, a particle's velocity is the time rate of change of its position. Furthermore, this velocity is tangent to the particle's trajectory at every position along its path. In a non-rotating frame of reference, the derivatives of the coordinate directions are not considered as their directions and magnitudes are constants.