The vector b can be expressed as:

where $b$ and $c$ are the components of the vector.

To elaborate, vectors are mathematical entities characterized by both magnitude (or length) and direction. In a two-dimensional space, we commonly utilize the unit vectors i and j to denote the directions along the x-axis and y-axis, respectively. The unit vector i points in the direction of the positive x-axis, while the unit vector j points in the direction of the positive y-axis.

To represent a vector b in terms of i and j, we need to identify its components along the x and y axes. For instance, if the vector b has an x-component of $b$ and a y-component of $c$, this indicates that b moves $b$ units along the x-axis and $c$ units along the y-axis.

We can express the vector b as a combination of these movements:

In this equation, the term $b \mathbf{i}$ represents the displacement along the x-axis, while $c \mathbf{j}$ denotes the displacement along the y-axis. By summing these two components, we obtain the complete vector b.

This representation of vectors enhances our ability to visualize and manipulate them, particularly when performing operations such as addition, subtraction, and scalar multiplication. Moreover, it simplifies the analysis of the vector’s direction and magnitude.