K323 / 2027
Physical quantities and measurement overview

Lesson 8 of 8

Adding vectors with a scale drawing

  • I can find the resultant of two vectors with a graphical method.Syllabus

    Syllabus K323, 1(g). Find the resultant of two vectors with a graphical method.

Make a guess. You are not marked.

Two forces of 3 N and 4 N act on the same point in any directions. Which value could NOT be their resultant?

  1. 1 N, because forces acting on one point cannot partly cancel.
  2. 5 N, because forces add to 7 N or subtract to 1 N, nothing in between.
  3. 8 N, because the largest possible resultant is 7 N.
Show the answer

8 N, because the largest possible resultant is 7 N.

The resultant is largest, 7 N, when the forces point the same way, and smallest, 1 N, when they are opposite. Any value between is possible; 8 N is not.

The resultant is the single vector equal to the sum of the vectors being added. A scale drawing lets you find its magnitude and direction.

A vector arrow represents both magnitude and direction. Keep both unchanged when moving an arrow for the construction.

First check the directions

Suppose two forces act at the same time on one object. If they are 3 N east and 2 N east, their resultant is 5 N east. If they are 3 N east and 2 N west, their resultant is 1 N east. Adding magnitudes alone would ignore the opposing direction.

For forces that are not along the same line, construct their vector sum head to tail. Draw the second arrow starting at the head of the first, without rotating it or changing its length. The resultant points from the first arrow's tail to the second arrow's head.

Moving the arrow is a drawing method for adding the vectors. In these examples, both forces act simultaneously on the same object; the arrows do not show forces applied one after the other.

Worked construction

3 N east and 4 N north

Two simultaneous forces act on one object: 3 N east and 4 N north. Find their resultant using a scale drawing.

  1. Choose a scale: on paper, use 1 cm to represent 1 N.
  2. Draw the first vector: draw AB 3 cm long towards the east.
  3. Draw the second vector: from B, draw BC 4 cm long towards the north. It makes a right angle with the east direction.
  4. Join the endpoints: draw the resultant arrow from A to C.
  5. Measure: AC is 5 cm long and makes an angle of about 53° north of east.
  6. Convert back: 5 cm represents 5 N, so the resultant is 5 N at about 53° north of east.
Head-to-tail addition of perpendicular forcesA 3 N east arrow runs from A to B. A 4 N north arrow runs from B to C. The resultant runs from A to C: 5 N at about 53 degrees north of east. These represent simultaneous forces on one object.ABC3 N east4 N northResultant5 N53°
AB represents 3 N east; BC represents 4 N north. AC is the resultant. On the stated paper scale, AB = 3 cm, BC = 4 cm and AC = 5 cm.

Construct the scale drawing on paper to measure it. The diagram on a screen changes size with the display, so its lengths need not be physical centimetres. Use the supplied lengths and scale. The method is construction and measurement, not remembering one particular triangle.

The method also works without a right angle

Now two simultaneous forces are 4 N east and 3 N at 60° north of east. The second force is diagonal, rather than directly north.

  1. Use the same paper scale, 1 cm per N. Draw AB 4 cm east.
  2. At B, draw a light reference line continuing east. Use a protractor to measure 60° towards the north from that line.
  3. Along that direction draw BC 3 cm long. Join A to C, with the arrowhead at C.
  4. Measure AC: about 6.1 cm. Its direction is about 25° north of east.
  5. The resultant is therefore about 6.1 N at 25° north of east.
Head-to-tail addition of forces at 60 degreesA 4 N east arrow runs from A to B. A 3 N arrow, at 60 degrees north of east, runs from B to C. The resultant from A to C is about 6.1 N at 25 degrees north of east.ABC4 N east3 N60°25°Resultantabout 6.1 NEastDirections measured north of east
The 60° angle is measured from east at B. The resultant's direction is measured from east at A. A careful paper construction gives approximately 6.1 cm and 25°; small drawing and reading differences are expected.

The answer is less than 4 + 3 = 7 N because the forces are not in the same direction. An arrow from C to A would have the correct length but the wrong direction for the resultant.

Check your understanding

Two force arrows are drawn head to tail: the first points from A to B and the second from B to C. Which arrow represents their resultant?

  1. A to C
  2. A to B
  3. C to A
Show the answer

A to C

The resultant joins the tail of the first vector to the head of the second. Moving an arrow for the construction is allowed only if its length and direction stay unchanged.

Practise an exam question

Write your answer first, then tick the mark points it makes.

Exam-style question

Two forces act on a point: 6.0 N towards the east and 8.0 N towards the north. Find the size and direction of the resultant force, either by calculation or by a scale drawing. [3]

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