Two vectors can be added together and the result of the vector sum is another vector. There are multiple methods to calculate the result of a vector sum, and which one to choose depends on the specific situation that needs to be addressed.

Method of the Pythagorean theorem

The Pythagorean theorem is a useful method for determining the result of adding two vectors at a right angle. The method is not applicable for adding more than two vectors or for adding vectors that are not at 90-degrees to each other. To add two vectors using the Pythagorean theorem, the vectors are placed forming the two short sides of a right triangle, and the magnitude of their vector sum can be calculated as the length of the hypotenuse of the triangle.

Head-to-tail method

A graphical method to add two or more vectors is the head-to-tail method. The head-to-tail method involves drawing a vector to scale beginning at a designated starting position. After we have drawn this first vector, we draw the vector we want to add to it with its tail on the head on the first vector. If we want to add an additional vector to these two, we draw it with its tail on the head of the last vector we drew. The process is repeated for all vectors that are being added. Once all the vectors have been added head-to-tail, the resultant is then drawn from the tail of the first vector to the head of the last vector.

Method of Cartesian coordinates

Vectors can be represented by a set of coordinates in a cartesian coordinate system. To add vectors given by their cartesian coordinates, we simply add each coordinate separately, taking into account the fact that some of the coordinates might be negative. The coordinates of the vector sum of two vectors are the sum of the corresponding coordinates of each vector.

For example, the vector sum of the vectors (1,2) and (-3, 4) has coordinates (-2, 6), because 1+ (-3)=-2 and 2+4=6. one two and minus three four has coordinates minus two six, because one plus minus three equals minus two and two plus four equals six.

Figure: The image shows three ways on how to sum vectors. The first method is the Pythagorean method, the second is using the Head-to-Tail method and the third is by the method of Cartesian coordinates.