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Calculating with Vectors (Cambridge O Level Physics)
Revision Note
Calculations with Vectors
- Vectors are represented by an arrow
- The arrowhead indicates the direction of the vector
- The length of the arrow represents the magnitude
The two force vectors acting on the object have both a direction and a magnitude
- Component vectors are sometimes drawn with a dotted line and a subscript indicating horizontal or vertical
- For example, Fv is the vertical component of the force F
Calculating Vectors Graphically
- Vectors at right angles to one another can be combined into one resultant vector
- The resultant vector will have the same effect as the two original ones
- To calculate vectors graphically means carefully producing a scale drawing with all lengths and angles correct
- This should be done using a sharp pencil, ruler and protractor
- Follow these steps to carry out calculations with vectors on graphs
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- Choose a scale which fits the page
- For example, use 1 cm = 10 m or 1 cm = 1 N, so that the diagram is around 10 cm high
- Draw the vectors at right angles to one another
- Complete the rectangle
- Draw the resultant vector diagonally from the origin
- Carefully measure the length of the resultant vector
- Use the scale factor to calculate the magnitude
- Use the protractor to measure the angle
- Choose a scale which fits the page
Vector drawn to scale
Vectors can be measured or calculated graphically if you are confident in using scales
Combining Vectors by Calculation
- In this method, a diagram is still essential but it does not need to be exactly to scale
- The diagram can take the form of a sketch, as long as the resultant, component and sides are clearly labelled
Resolving vectors
Resolving two force vectors F1 and F2 into a resultant force vector FR
Resultant vector
- Use Pythagoras' Theorem to find the resultant vector
Pythagoras' Theorem
Pythagoras's Theorem makes calculating vectors at right angles much simpler
- Use trigonometry to find the angle
- The mnemonic 'soh-cah-toa' is used to remember how to apply sines and cosines to resolve the sides of a triangle
Trigonometry and Pythagoras' Therom are essential in vector calculations
Worked example
A hiker walks a distance of 6 km due east and 10 km due north.
Calculate the magnitude of their displacement and its direction from the horizontal.
Answer:
Step 1: Draw a vector diagram
Step 2: Calculate the magnitude of the resultant vector using Pythagoras' Theorem
Resultant vector = 11.66
Step 3: Calculate the direction of the resultant vector using trigonometry
Step 4: State the final answer complete with direction
Resultant vector = 12 km 59° east and upwards from the horizontal
Examiner Tip
If the question specifically asks you to use the calculation or graphical method, you must solve the problem as asked. However, if the choice is left up to you then any correct method will lead to the correct answer.
The graphical method sometimes feels easier than calculating, but once you are confident with trigonometry and Pythagoras you will find calculating quicker and more accurate.
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