Defining the Position of a Point

In two dimensions (2D), a point can be defined using an (x, y) coordinate pair.
In this system, ‘x’ denotes the horizontal position and ‘y’ denotes the vertical position of the point.
This system is called the cartesian coordinate system, and it consists of two perpendicular axes (the x-axis and the y-axis).
The point where the x-axis and y-axis intersect is called the origin, represented as (0,0).
The position of a point is described relative to this origin.

In 3D space, a point is represented by three coordinates (x, y, z), where ‘x’ and ‘y’ represent the horizontal axes and ‘z’ represents the vertical axis.
A point’s magnitude, or distance from the origin, is calculated using Pythagoras’ theorem.
In 2D, the distance of point (x, y) from the origin (0, 0) is given by sqrt(x² + y²).
In 3D, the distance of point (x, y, z) from the origin (0, 0, 0) is given by sqrt(x² + y² + z²).

In physics and engineering, it’s more common to use vectors to define the position of a point.
A vector is a quantity that specifies both a direction and a magnitude.
So, a point’s position can be defined as a vector from the origin to that point.
This vector is represented as (x, y) in 2D, or as (x, y, z) in 3D.
The magnitude of a vector (its length) can be calculated in the same way as the distance from the origin.

Translating a point means moving it a certain distance in a certain direction.
This can be done by adding a translation vector to the point’s position vector.
For example, translating a 2D point (x, y) by the vector (a, b) results in the point (x + a, y + b).
And translating a 3D point (x, y, z) by the vector (a, b, c) results in the point (x + a, y + b, z + c).