Addition of Vectors using the components method

When adding two or more vectors using the components method follow these guidelines.

  1. Sketch the vectors on a piece of paper.  This will give you an idea of their relative positions and magnitudes.
  2. Calculate each vector's horizontal (x) and vertical component (y) using trigonometry
  3. add all the horizontal components together (taking into account the sign of each vector component)
  4. add all the vertical components together (taking into account the sign o feach vector component)
  5. Now you have a resultant horizontal vector in the x direction and a resultant vertical vector in the y direction.
  6. Use the resultant x and y vectors to calculate the total resultant vector.

  7. Calculate the angle and state the direction of the final (resultant) vector.

This animation illustrates that, irrespective of the path, from the initial position (the tail of first vector) to the final position (the head of last vector), the resultant vector VTot will have only two components: an (horizontal) component and a y (vertical) component.  Here the vectors ab, and c each have their individual x and y components which, when added together to all the x and y components respectively, will give a total horizontal component V1x and a total vertical component V2y. 

Find the final displacement of a man who first walks North for 5.0 km, then East for 3.0 km and finally [E600S] for 8.0 km.  

Given: d1 = 5.0 km [N],  d2 = 3.0 km [E], d= 8.0 km [E600S]


  • All components
  •  dtx
  •  dty
  •  dt
  •  theta (the direction of the final displacement vector dt)

noteHere is a step-by-step method to approach the solution of problems involving the addition of vectors using the components method.

  1. Draw a quick sketch to guide you through the solution (doesn't have to be to scale - this is not a graphical solution)

    comp exampfin
  2. Set-up a table for each vector where you can enter the x and y components of each.
Identify each 
Draw a quick diagram 
for each vector for reference
calculate the
using trigonometry
calculate the
using trigonometry
d1 = 5.0 km [N] comp exampd1  d1x = 0  d1y = +5
d2 = 3.0 km [E] comp examp3  d2x = + 3  d2y = 0
d3 = 8.0 km [E600S] comp examp6  vector14  vector15
( x and y components together)
comp examp  dTx = 0 + 3 + 4 = +7 km 
[to the right]
 dTy = 5 + 0+ (-)6.9
 = -1.9 km [down]


3. Once we have the components for the resultant dTx and dTy we can find the direction using inverse trigonometric functions

noteRecall  from your trigonometry 

dTy/dTx = tan theta

Therefore: theta = inv tan (dTy/dTx)= 150

4. Use Pythagoras' Theorem to find the magnitude of total displacement :


5. One final statement indicating the final answer with both magnitude and direction

 The the man's total displacement is : 7.3 km [E150]