Motion in Three Dimensions

A particle moving in a 1-dimensional space requires only one coordinate to specify its position. In two dimensions similarly, two coordinates are required. Three-dimensional motions are encountered at a lot of places in real life, to analyze these real-life situations. One needs to understand the motion and how to deal with these three coordinates mathematically to describe the trajectories of objects traveling in a three-dimensional plane. Let's look at these concepts in detail.

Motion in a Three-Dimensional Space

Suppose a particle is moving between two points in a three-dimensional space. To describe the position of this particle, a position vector is required. These vectors are always with respect to the reference frame at the origin. The following parameters are required to fully describe the behavior of a particle moving in a plane,

1. Position
2. Velocity
3. Acceleration

Position Vector

In a 3-d space, a particle can be anywhere, it cannot be described by just one coordinate. In this case, it is denoted with respect to the origin and it also constitutes the direction one should go in to find that point. That is why a vector is required to describe the position. The vector which denotes the position and direction of the particle's position with respect to the origin is called the position vector. The position vector \vec{r}  for a particle is given by, 

\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}

Where x,  y, and z are their components along the x, y, and z-axis. 

Velocity

The velocity of a particle traveling in the three-dimensional space can be described in two ways - average velocity and instantaneous velocity. When the particle is under acceleration, it changes its velocity every second. So, a single value cannot be assigned to a velocity. In such cases, the instantaneous velocity is preferred, it describes the velocity and its direction at a particular instant. It is given by,

\vec{v} = \lim_{\Delta t \to 0}\frac{\Delta \vec{r}}{\Delta t}\\ = \vec{v} = \frac{dr}{dt}

Velocity can also be expressed as, 

v = \frac{dx}{dt}\hat{i} + \frac{dy}{dt}\hat{j} + \frac{dz}{dt}\hat{k}

The average velocity is the ratio of total displacement over total time. Suppose a particle goes from  \vec{r}  to \vec{r'}  in a total time of \Delta t

The velocity is given by, 

\vec{v} = \frac{\vec{r} - \vec{r'}}{\Delta t}

Acceleration

The acceleration of a body moving in a plane is given b