Calculating with Vectors (CIE IGCSE Physics)

EXTENDED

• 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, F_v 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

1. 1. 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
   2. Draw the vectors at right angles to one another
   3. Complete the rectangle
   4. Draw the resultant vector diagonally from the origin
   5. Carefully measure the length of the resultant vector
   6. Use the scale factor to calculate the magnitude
   7. Use the protractor to measure the angle

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 two force vectors F₁ and F₂ into a resultant force vector F_R

• Use Pythagoras' Theorem to find the resultant vector

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

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