2.3:
Vector Components in the Cartesian Coordinate System
The Cartesian coordinate system consists of three mutually perpendicular axes defined by unit vectors intersecting at the origin.
Unit vectors have unit magnitude and only point the direction.
For Cartesian systems, î, ĵ, and k̂ are the unit vectors along the positive x, y, and z axes, respectively.
In this frame, every vector is the sum of its orthogonal projections onto each of the axes. These projections are known as the vector components.
We write each vector component by its magnitude and a unit vector along the axis. The magnitudes represent the scalar components of the vector.
Thus, any vector is the vector sum of its components.
Using the scalar components, the magnitude of a vector is given by the square root of the sum of the squares of its components.
For a vector on a plane, its direction, which is the angle made by the vector with the positive x-axis in the anticlockwise direction, is the tan inverse of the y-component over the x-component.
Conversely, the x-component of a vector is the magnitude times the cosine of the angle made with the x-axis.
Its y-component is the magnitude times the sine of the angle made by the vector with the x-axis.
2.3:
Vector Components in the Cartesian Coordinate System
Vectors are usually described in terms of their components in a coordinate system. Even in everyday life, we naturally invoke the concept of orthogonal projections in a rectangular coordinate system. For example, if someone gives you directions for a particular location, you will be told to go a few km in a direction like east, west, north, or south, along with the angle in which you are supposed to move. In a rectangular (Cartesian) xy-coordinate system in a plane, a point in a plane is described by a pair of coordinates (x, y). In a similar fashion, a vector in a plane is described by a pair of its vector coordinates. The x-coordinate of a vector is called its x-component, and the y-coordinate is called its y-component. In the Cartesian system, the x and y vector components are the orthogonal projections of this vector onto the x– and y-axes, respectively. In this way, each vector on a Cartesian plane can be expressed as the vector sum of its vector components in both x and y directions.
It is customary to denote the positive direction of the coordinate axes by unit vectors. The vector components can now be written as their magnitude multiplied by the unit vector in that direction. The magnitudes are considered as the scalar components of a vector.
When we know the scalar components of any vector, we can find its magnitude and its direction angle. The direction angle, or direction for short, is the angle the vector forms with the positive direction on the x-axis. The angle that defines any vector's direction is measured in a counterclockwise direction from the +x-axis to the vector. The direction angle of any vector is defined via the tangent function. It is defined as the ratio of the scalar y component to the scalar x component of that vector.
In many applications, the magnitudes and directions of vector quantities are known, and we need to find the resultant of many vectors. In such cases, we find vector components from the direction and magnitude. Thus, the x-component is given by the product of the magnitude of that vector and the cosine angle made with the x-axis in the counterclockwise direction. Similarly, the y-component is the product of vector magnitude and the sine angle made with the x-axis in the counterclockwise direction.
This text is adapted from Openstax, University Physics Volume 1, Section 2.2: Coordinate Systems and Components of a Vector.