Mechanics

# 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.

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  .