1.6 Adding Vectors

• Vectors can be combined by adding or subtracting them to produce the resultant vector
  • The resultant vector is sometimes known as the ‘net’ vector (eg. the net force)

• There are two methods that can be used to add vectors
  • Calculation – if the vectors are perpendicular
  • Scale drawing – if the vectors are not perpendicular

Combining Vectors Using a Scale Diagram

• There are two methods that can be used to combine vectors using a scale diagram: the triangle method and the parallelogram method

• To combine vectors using the triangle method:
  • Step 1: link the vectors head-to-tail
  • Step 2: the resultant vector is formed by connecting the tail of the first vector to the head of the second vector

• To combine vectors using the parallelogram method:
  • Step 1: link the vectors tail-to-tail
  • Step 2: complete the resulting parallelogram
  • Step 3: the resultant vector is the diagonal of the parallelogram

Combining Vectors by Calculating

• Combining vectors by calculation is a two-step process
  • Finding the direction of the resultant using trigonometry
  • Finding the magnitude of the resultant using Pythagoras

The magnitude of the resultant vector is found by using Pythagoras’ Theorem

• The direction of the resultant vector is found from the angle it makes with the horizontal or vertical
  • The question should imply which angle it is referring to (ie. Calculate the angle from the x-axis)

• Calculating the angle of this resultant vector from the horizontal or vertical can be done using trigonometry
  • Either the sine, cosine or tangent formula can be used depending on which vector magnitudes are calculated

The direction of vectors is found by using trigonometry

Step 1: Sketch a diagram, including all known values

Step 2: Calculate the magnitude of the resultant using Pythagoras

= 5.4 

Step 3: Calculate the angle using trigonometry

θ = 21.8

Step 4: Write the answer in full giving both magnitude and direction of the velocity and all units

The swimmer's velocity is 5.4 ms^-1 at 22^o to the horizontal direction