Many physical quantities we deal with are represented as vector quantities, such as velocity, force etc. These quantities interact with each other and produce a resultant effect on the objects upon which they are applied. Since the impact of all these forces is taken into consideration when finding the nature of motion of the system, so, in order to find the resultant of these forces, operations such as addition, subtraction and multiplication are required to be performed on these forces. In this section, we will learn about the vector addition of two quantities using the analytical methods.

We all have learned about the graphical method of addition of two vector quantities with the triangle law of vector addition or the parallelogram law of vector addition. We know it’s a tedious process and the accuracy of this process is not reliable. In this section, we will learn about the analytical method of vector addition. Let us consider two vectors, represented as shown below,

Let R be the sum of these vectors. R can be given as,

extended to addition and subtraction of two or more vectors. Let us illustrate this with the following example.

Let us take three vectors as shown below,

where i is the unit vector along the x-axis, j is the unit vector along the y-axis and j is the unit vector along the z-axis. Let us understand the operations of addition and subtraction in vectors shown with the following example.

Example: Perform the following vector operation on the vectors A, B and C.

Solution: The above operation can be performed on these vectors with the following method.

Where the vector  components of the vector T is given by,