For AI agents: use /llms.txt for the Nakafa content index.
To specify the position of an object, such as an aircraft in flight, we need a frame of reference or coordinate system. The most commonly used system is the Cartesian Coordinate System, developed by René Descartes.
The Cartesian Coordinate System uses perpendicular axes to determine the location of a point.
In two dimensions, we use the axis (horizontal) and axis (vertical). A point's position is expressed as an ordered pair , for example, point .
This means that point is located to the right (positive axis) and upward (positive axis) from the origin .
In three dimensions, we add a axis that is perpendicular to the plane. A point's position is expressed as , used for determining the position of objects like aircraft in space.
We can think of as length, as width, and as height.
Unlike coordinates which represent the position of a point, a vector is a quantity that has both magnitude (length) and direction. Imagine a fly traveling from point to point . The fly's displacement can be represented as a vector.
Position Vector: A vector starting from the origin to a point . The position vector of is written as .
Example: .
Displacement Vector (or Free Vector): A vector connecting any two points, for example, from point to point . This vector is written as . It doesn't have to start from the origin.
Vectors can be written in several ways:
Row Vector: Using regular parentheses.
Example: .
Column Vector: Using square brackets or large parentheses.
Example: .
Unit Vector Notation for Three Dimensions: Using a combination of unit vectors , , and which are aligned with the , , and axes.
Imagine a map application on your phone. This application relies on coordinate systems to:
Without coordinate systems, it would be impossible for applications to accurately understand position and direction. Every movement, such as an aircraft changing altitude or direction, needs to be reported and processed using coordinate systems and vector concepts.
Example: displacement vectors from to or from to .
Example: a vector can be written as , meaning vector has a component of in the direction, in the direction, and in the direction.
