# vector operations

Vector with horizontal direction and direction from left to right

You are already used to working with scalar quantities and therefore know that they add up according to the common rules of algebra. For example: if you have an area of 400 m ^{2} and you buy the neighboring land, with an area of 500 m ^{2} , your area will now be 900 m ^{2} .

We must pay close attention when operating with vectors, as the operation mechanism is different from the operation with numbers, since it does not only involve numerical values, but also spatial orientations. Therefore, the rules for vector algebra are different from those used for number algebra.

**Adding vectors**

We can perform mathematical operations such as addition and subtraction of vectors. Consider two vectors and , represented by the segments shown in the figure below.

The **sum** vector or **resultant vector** ( ) of the two mentioned vectors, such that

can be obtained, in general, with the help of the polygon rule, which is a graphical method. Let’s represent the vector of the above operation following the polygon rule step by step.

**polygon rule**

To get , we use the following process:

– first we draw the representative segment of the vector using any point on the plane as origin;

– later we draw the representative segment of the vector , so that its origin coincides with the end of the vector ;

– and, finally, the sum vector will be represented by the oriented segment whose origin coincides with the vector and whose end coincides with the vector .

To determine the sum vector or the resultant vector S of two other vectors (for example, vectors and ) it is necessary to trace the vector so that its origin coincides with the end of the other vector, in this case, the vector .

Therefore, we find the resulting vector when we join the vector’s origin to the vector ‘s endpoint .