For AI agents: use /llms.txt for the Nakafa content index.
A slanted vector is difficult to calculate with if we only see it as one arrow. To make calculations easier, the vector is split into parts parallel to the coordinate axes. These parts are called vector components.
The horizontal component shows how much of the vector lies along the axis. The vertical component shows how much lies along the axis. In three-dimensional space, there is also a component along the axis.
Vector components turn one slanted arrow into parts that follow the axes.
In a plane, vector can be written as the combination of its and components.
The notation is the unit vector in the direction, while is the unit vector in the direction. The numbers and give the component sizes along each direction.
For example, means the vector moves units in the positive direction and units in the positive direction.
The vector length comes from the Pythagorean theorem because the and components are perpendicular.
In three-dimensional space, a vector needs three components. The general form is:
The unit vector points along the axis. This component is needed when motion or force is not limited to a flat plane, such as a drone moving east, north, and upward at the same time.
The visual uses component steps in order: move along the axis, then the axis, then the axis. The final endpoint is the same as the endpoint of vector .
Components can be positive or negative. The sign shows direction relative to the reference axis, not a negative vector magnitude.
| Component | Positive sign means | Negative sign means |
|---|---|---|
| toward positive | toward negative | |
| toward positive | toward negative | |
| toward positive | toward negative |
If , the vector has a unit component toward negative and a unit component toward positive .
A student walks east, then north. If east is the positive direction and north is the positive direction, the displacement is:
Components keep the calculation organized because each direction is handled on its own axis.
