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Techniques of vector addition

Module by: Free High School Science Texts Project. E-mail the author

Techniques of Vector Addition

Now that you have learned about the mathematical properties of vectors, we return to vector addition in more detail. There are a number of techniques of vector addition. These techniques fall into two main categories - graphical and algebraic techniques.

Graphical Techniques

Graphical techniques involve drawing accurate scale diagrams to denote individual vectors and their resultants. We next discuss the two primary graphical techniques, the head-to-tail technique and the parallelogram method.

The Head-to-Tail Method

In describing the mathematical properties of vectors we used displacements and the head-to-tail graphical method of vector addition as an illustration. The head-to-tail method of graphically adding vectors is a standard method that must be understood.

Method: Head-to-Tail Method of Vector Addition

  1. Draw a rough sketch of the situation.
  2. Choose a scale and include a reference direction.
  3. Choose any of the vectors and draw it as an arrow in the correct direction and of the correct length – remember to put an arrowhead on the end to denote its direction.
  4. Take the next vector and draw it as an arrow starting from the arrowhead of the first vector in the correct direction and of the correct length.
  5. Continue until you have drawn each vector – each time starting from the head of the previous vector. In this way, the vectors to be added are drawn one after the other head-to-tail.
  6. The resultant is then the vector drawn from the tail of the first vector to the head of the last. Its magnitude can be determined from the length of its arrow using the scale. Its direction too can be determined from the scale diagram.
Exercise 1: Head-to-Tail Addition I

A ship leaves harbour H and sails 6 km north to port A. From here the ship travels 12 km east to port B, before sailing 5,5 km south-west to port C. Determine the ship's resultant displacement using the head-to-tail technique of vector addition.

Solution
  1. Step 1. Draw a rough sketch of the situation :

    Its easy to understand the problem if we first draw a quick sketch. The rough sketch should include all of the information given in the problem. All of the magnitudes of the displacements are shown and a compass has been included as a reference direction. In a rough sketch one is interested in the approximate shape of the vector diagram.

    Figure 1
    Figure 1 (PG11C1_040.png)

  2. Step 2. Choose a scale and include a reference direction :

    The choice of scale depends on the actual question – you should choose a scale such that your vector diagram fits the page.

    It is clear from the rough sketch that choosing a scale where 1 cm represents 2 km (scale: 1 cm = 2 km) would be a good choice in this problem. The diagram will then take up a good fraction of an A4 page. We now start the accurate construction.

  3. Step 3. Choose any of the vectors to be summed and draw it as an arrow in the correct direction and of the correct length – remember to put an arrowhead on the end to denote its direction :

    Starting at the harbour H we draw the first vector 3 cm long in the direction north.

    Figure 2
    Figure 2 (PG11C1_041.png)

  4. Step 4. Draw the second vector:

    Take the next vector and draw it as an arrow starting from the head of the first vector in the correct direction and of the correct length.

    Since the ship is now at port A we draw the second vector 6 cm long starting from point A in the direction east.

    Figure 3
    Figure 3 (PG11C1_042.png)

  5. Step 5. Draw the third vector:

    Take the next vector and draw it as an arrow starting from the head of the second vector in the correct direction and of the correct length.

    Since the ship is now at port B we draw the third vector 2,25 cm long starting from this point in the direction south-west. A protractor is required to measure the angle of 45.

    Figure 4
    Figure 4 (PG11C1_043.png)

  6. Step 6. Draw the resultant:

    The resultant is then the vector drawn from the tail of the first vector to the head of the last. Its magnitude can be determined from the length of its arrow using the scale. Its direction too can be determined from the scale diagram.

    As a final step we draw the resultant displacement from the starting point (the harbour H) to the end point (port C). We use a ruler to measure the length of this arrow and a protractor to determine its direction.

    Figure 5
    Figure 5 (PG11C1_044.png)

  7. Step 7. Apply the scale conversion :

    We now use the scale to convert the length of the resultant in the scale diagram to the actual displacement in the problem. Since we have chosen a scale of 1 cm = 2 km in this problem the resultant has a magnitude of 9,2 km. The direction can be specified in terms of the angle measured either as 072,3072,3 east of north or on a bearing of 072,3072,3.

  8. Step 8. Quote the final answer :

    The resultant displacement of the ship is 9,2 km on a bearing of 072,3072,3.

Exercise 2: Head-to-Tail Graphical Addition II

A man walks 40 m East, then 30 m North.

  1. What was the total distance he walked?
  2. What is his resultant displacement?
Solution
  1. Step 1. Draw a rough sketch :

    Figure 6
    Figure 6 (PG11C1_045.png)

  2. Step 2. Determine the distance that the man traveled :

    In the first part of his journey he traveled 40 m and in the second part he traveled 30 m. This gives us a total distance traveled of 40 m + 30 m = 70 m.

  3. Step 3. Determine his resultant displacement :

    The man's resultant displacement is the vector from where he started to where he ended. It is the vector sum of his two separate displacements. We will use the head-to-tail method of accurate construction to find this vector.

  4. Step 4. Choose a suitable scale :

    A scale of 1 cm represents 10 m (1 cm = 10 m) is a good choice here. Now we can begin the process of construction.

  5. Step 5. Draw the first vector to scale :

    We draw the first displacement as an arrow 4 cm long in an eastwards direction.

    Figure 7
    Figure 7 (PG11C1_046.png)

  6. Step 6. Draw the second vector to scale :

    Starting from the head of the first vector we draw the second vector as an arrow 3 cm long in a northerly direction.

    Figure 8
    Figure 8 (PG11C1_047.png)

  7. Step 7. Determine the resultant vector :

    Now we connect the starting point to the end point and measure the length and direction of this arrow (the resultant).

    Figure 9
    Figure 9 (PG11C1_048.png)

  8. Step 8. Find the direction :

    To find the direction you measure the angle between the resultant and the 40 m vector. You should get about 37.

  9. Step 9. Apply the scale conversion :

    Finally we use the scale to convert the length of the resultant in the scale diagram to the actual magnitude of the resultant displacement. According to the chosen scale 1 cm = 10 m. Therefore 5 cm represents 50 m. The resultant displacement is then 50 m 3737 north of east.

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