Figure 1 Coordinate system

Component of a vector
Vectors are usually added analytically or mathematically. In order to do that, we need to define the components of a vector. Let’s consider a component of any vector.

Figure 2.8

We are given a vector . We place it in the Cartesian (rectangular) coordinate system, as shown in Figure 2.8. The tail of the vector is at 0, the origin of the coordinate system. From Figure 2.8 we see that the horizontal component vector, , that lies on or is parallel to the x-axis is called the x-component. The vertical component vector, , that lies on or is parallel to the y-axis is called the y-component (Fig.2.8). Sum of these two components is equal to .

Figure 2.9

Figure 2.10

Figure2.11

The x- and y-components of vectors can also be expressed as signed numbers. The absolute value of the signed number corresponds to the magnitude (length) of the component vector.
When the component vector points in the positive x-direction, we define the number to be equal to the magnitude of . We define the number in the same way. The two numbers and are called the components of .
When the component vector points in the negative x-direction, we define the number to be equal to the negative of that magnitude (the magnitude of a vector quantity is itself never negative). The sign of the number corresponds to the direction of the component as follows: