Adding Vectors

You should be able to think of at least four vector quantities (size plus direction) and four scalar quantities (size only). If you can’t, look here.

You can’t just add numerically when you want to add two or more vectors (of the same kind, obviously) since a push of 1N to the left when added to a pull of 3.5N to the right results in a pull of 2.5N to the right. We have to take direction into account.

But, what if they don’t act along the same straight line? There are several methods for adding lots of them together. The head-to-tail method is one. A vector is just a line on a piece of paper of a particular length which represents its size with an arrow on it to indicate direction.  Adding two vectors A and B is quite simple. Take the tail of B and put it on the head of A. The vector sum is found by the line joining the tail of A and the head of B. Works for as many vectors as you like.

In more detail, 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 angle of rotation – in my example anticlockwise from due East, but clockwise from North can also be used. Just specify.

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 using a ruler and protractor. Label the magnitude and direction of the scale on the diagram (e.g., SCALE: 1 cm = 20N for adding forces, for example).
  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 20N per cm = 88N. Here’s an example:


Add these three vectors together. Ask yourself – does the order matter? Think about it…