A variety of mathematical operations can be performed with and upon vectors. One such operation is the addition of vectors. Two vectors can be added together to determine the result (or resultant). The rules for summing vectors (such as force vectors) were kept relatively simple. Observe the following summations of two force vectors:

The head-to-tail method involves drawing a vector to scale on a sheet of paper beginning at a designated starting position. Where the head of this first vector ends, the tail of the second vector begins (thus, head-to-tail method). 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; i.e., from start to finish. Once the resultant is drawn, its length can be measured and converted to real units using the given scale. The direction of the resultant can be determined by using a protractor and measuring its counterclockwise angle of rotation from due East.

A step-by-step method for applying the head-to-tail method to determine the sum of two or more vectors is given below.

1. Choose a scale and indicate it on a sheet of paper. The best choice of scale is one that will result in a diagram that is as large as possible, yet fits on the sheet of paper.
2. Pick a starting location and draw the first vector to scale in the indicated direction. Label the magnitude and direction of the scale on the diagram (e.g., SCALE: 1 cm = 20 m).
3. Starting from where the head of the first vector ends, draw the second vector to scale in the indicated direction. Label the magnitude and direction of this vector on the diagram.
4. Repeat steps 2 and 3 for all vectors that are to be added
5. Draw the resultant from the tail of the first vector to the head of the last vector. Label this vector as Resultant or simply R.
6. Using a ruler, measure the length of the resultant and determine its magnitude by converting to real units using the scale (4.4 cm x 20 m/1 cm = 88 m).
7. Measure the direction of the resultant using the counterclockwise convention.

An example of the use of the head-to-tail method is illustrated below. The problem involves the addition of three vectors:

### SCALE: 1 cm = 5 m

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To add vectors mathematically we first must split each vector into two separate vectors, each of these new vectors will lie entirely on one of the two (or three if the vector is a in 3 dimensions) axis. Normally we call these axis x and y, with the x on the horizontal axis and the y is the vertical axis.

Finding the components of vectors for vector addition involves forming a right triangle from each vector and using the standard triangle trigonometry. Take the two vectors below.

After finding the components for the vectors A and B, and combining them to find the components of the resultant vector R, the result can be put in component form.

Example of the head to tail method

Examples using the mathematical method

Use this link to solve the following questions

PHeT