Why Vectors Are Split into Components
A slanted vector acts along more than one coordinate axis. We split it into parts parallel to those axes so that the effect in each direction can be calculated separately. These parts are called .
For AI agents: use /llms.txt for the Nakafa content index.
A slanted vector acts along more than one coordinate axis. We split it into parts parallel to those axes so that the effect in each direction can be calculated separately. These parts are called .
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.
A vector equals the sum of its components along the coordinate 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 signed scalars and give the components along those axes. Each sign records which way along its axis the vector points.
For example, means the vector has a unit component in the positive direction and an unit component 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 the direction relative to the reference axis. The vector's magnitude stays zero or positive.
| 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:
With components, the horizontal and vertical directions are calculated on their own axes.
The component form shows a net displacement toward the east and north. Its magnitude comes from a right triangle with side lengths and .
Published: . Updated: .