Calculating with Vectors

Exam code: 5054

Written by: Ashika|Reviewed by: Caroline Carroll|Updated 2 July 2026

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, F_v is the vertical component of the force F

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

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

Vectors can be measured or calculated graphically if you are confident in using scales

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

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

$Resultant vector = \sqrt{6^{2} + 10^{2}}$

$Resultant vector = \sqrt{136}$

Resultant vector = 11.66

Step 3: Calculate the direction of the resultant vector using trigonometry

$\tan θ = \frac{opposite}{adjacent} = \frac{10}{6}$

$θ = \tan^{- 1} (\frac{10}{6}) = 59 °$

Step 4: State the final answer complete with direction

Resultant vector = 12 km 59° east and upwards from the horizontal

Calculations with Vectors