Scale Diagrams
- There are two methods that can be used to combine or resolve vectors
- Calculation – if the vectors are perpendicular
- Scale drawing – if the vectors are not perpendicular
- Calculating vectors using a scale drawing involves drawings the lengths and angles of the vectors accurately using a sharp pencil, ruler and protractor
Vectors can be determined using scale diagrams
- When two vectors are not at right angles, the resultant vector can be calculated using a scale drawing
- Step 1: Link the vectors head-to-tail if they aren’t already
- Step 2: Draw the resultant vector using the triangle or parallelogram method
- Step 3: Measure the length of the resultant vector using a ruler
- Step 4: Measure the angle of the resultant vector (from North if it is a bearing) using a protractor
A scale drawing of two vector additions. The magnitude of resultant vector R is found using a rule and its direction is found using a protractor
- Note that with scale drawings, a scale may be given for the diagram such as 1 cm = 1 km since only limited lengths can be measured using a ruler
- The final answer is always converted back to the units needed in the diagram
- Eg. For a scale of 1 cm = 2 km, a resultant vector with a length of 5 cm measured on your ruler is actually 10 km in the scenario
Worked example
A hiker walks a distance of 6 km due east and 10 km due north.
By making a scale drawing of their route, find the magnitude of their displacement and its direction from the horizontal.
Answer:
Step 1: Choose a sensible scale
- The distances are 6 and 10 km, so a scale of 1 cm = 1 km will fit easily on the page, but be large enough for an accurate scale drawing
Step 2: Draw the two components using a ruler and make the measurements accurate to 1 mm
Step 3: Add the resultant vector, remembering the start and finish points of the journey
Step 4: Carefully measure the length of the resultant and convert using the scale
Step 5: Measure the angle between the vector and the horizontal line
Step 6: Write the complete answer, giving both magnitude and direction
- Magnitude: R = 11.7 km
- Direction: θ = 59°
Exam Tip
It should be noted that some of the examples used on this page demonstrate the use of scale diagrams where the vectors are placed at right angles - it would be quicker to determine the resultant force of these via calculation as simple trigonometry can be used
Scale diagram questions will typically involve vector triangles that do not contain a right angle
