Motion in more than one dimension requires position, velocity and acceleration to carry direction as well as magnitude.

Let the position vector be

$$\mathbf r(t)=(x(t),y(t),z(t)).$$

Velocity

The velocity vector is the derivative of position:

$$\mathbf v(t)=\frac{d\mathbf r}{dt} =\left(\frac{dx}{dt},\frac{dy}{dt},\frac{dz}{dt}\right).$$

It points in the instantaneous direction of motion and is tangent to the trajectory.

Acceleration

Acceleration is the derivative of velocity:

$$\mathbf a(t)=\frac{d\mathbf v}{dt} =\frac{d^2\mathbf r}{dt^2}.$$

Each component can be calculated independently.

Direction matters

Acceleration need not point in the same direction as velocity. A component parallel to velocity changes speed, while a perpendicular component changes the direction of motion.

An object can therefore accelerate even while moving at constant speed, as occurs in circular motion.

Component form

Vector kinematics converts a multidimensional motion problem into simultaneous one-dimensional problems along the coordinate axes, while the vector description preserves the geometry connecting those components.