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.