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Calculating with Vectors (CIE IGCSE Physics)
Revision Note
Calculations with vectors
Extended tier only
- Vectors can be drawn using vector diagrams
Vector diagrams
- Vectors are represented by an arrow
- The length of the arrow represents the magnitude
- The direction of the arrow indicates the direction
- the scale of the arrows should be proportional to the relative magnitudes of the forces
- an arrow for a 4 N force should be twice as long as an arrow for a 2 N force
Vector diagram of two forces acting on an object
The length of the arrows are proportional to the magnitude of the forces, and show the direction that forces act in
Calculating vectors graphically
- Vector diagrams can be used to combine vectors
- Vectors at right angles to one another can be combined into one resultant vector
- The resultant vector will have the combined effect of the two original vectors
- For example, a resultant force vector will have the combined effect of two component forces
- Component vectors are sometimes drawn with a dotted line and a subscript indicating horizontal or vertical
- A force , for example, may have two components:
- is the vertical component of the force
- is the horizontal component of force
- A force , for example, may have two components:
- 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
Vectors can be measured or calculated graphically using scaled vector diagrams
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Combining vectors by calculation
- In this method, a vector diagram is still essential but it does not need to be exactly to scale
- The vector diagram can take the form of a sketch, as long as the resultant side, component sides are clearly labelled
Using a vector diagram to resolve two force vectors F1 and F2 into a resultant force vector FR
- When the magnitude of only one vector is known, and the angle is known, then trigonometry can be used to find the magnitude of the missing vector
- The mnemonic 'soh-cah-toa' can used to remember the trigonometric functions
Trigonometry can be used when the magnitude of one vector and the angle is known
- When the magnitudes of two of the vectors are known, then Pythagoras' theorem can be used to find the magnitude of the missing vector
Pythagoras's theorem can be used when the magnitudes of two of the three vectors are known
Worked example
A force acts on an object with 60 N to the left. A second force of 100 N acts on the same object in the upward direction.
Calculate the resultant force acting on the object.
Answer:
Step 1: Draw a vector diagram
Step 2: Calculate the magnitude of the resultant force using Pythagoras' theorem
Step 3: Calculate the direction of the resultant vector using trigonometry
Step 4: State the final answer, complete with magnitude and direction
Exam 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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