# 2.3.2 Combining Vectors

### Combining Vectors

• Vectors are represented by an arrow
• The arrowhead indicates the direction of the vector
• The length of the arrow represents the magnitude
• Vectors can be combined by adding or subtrating 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

#### Vector Calculation

• Vector calculations will be limited to two vectors at right angles
• This means the combined vectors produce a right-angled triangle and the magnitude (length) of the resultant vector is found using Pythagoras’ theorem 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

#### Scale Drawing

• When two vectors are not at right angles, the resultant vector can be calculated using a scale drawing
• Step 2: Draw the resultant vector using the triangle or parallelogram method
• Step 3: Measure the length of the resultant vector using a ruler
• Step 4: Measure the angle of the resultant vector (from North if it is a bearing) using a protractor A scale drawing of two vector additions. The magnitude of resultant vector R is found using a rule and its direction is found using a protractor

• Note that with scale drawings, a scale may be given for the diagram such as 1 cm = 1 km since only limited lengths can be measured using a ruler
• The final answer is always converted back to the units needed in the diagram
• Eg. For a scale of 1 cm = 2 km, a resultant vector with a length of 5 cm measured on your ruler is actually 10 km in the scenario
• There are two methods that can be used to combine vectors: the triangle method and the parallelogram method
• To combine vectors using the triangle method:
• 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

#### 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 #### Exam Tip

Pythagoras’ Theorem and trigonometry are consistently used in vector addition, so make sure you’re fully confident with the maths here!

### Combining Vectors with a Vector Triangle

• Coplanar forces can be represented by the resultant of any two vector triangles
• Forces are in equilibrium if an object is either
• At rest
• Moving at constant velocity
• In equilibrium, coplanar forces are represented by closed vector triangles
• The vectors, when joined together, form a closed path
• The most common forces on objects are
• Weight
• Normal reaction force
• Tension (from cords and strings)
• Friction
• The forces on a body in equilibrium are demonstrated below:

#### Worked Example

A weight hangs in equilibrium from a cable at point X. The tensions in the cables are T1 and T2 as shown. Which diagram correctly represents the forces acting at point X?  #### Exam Tip

The diagrams in exam questions about this topic tend to be drawn to scale, so make sure you have a ruler handy! ### Author: Ashika

Ashika graduated with a first-class Physics degree from Manchester University and, having worked as a software engineer, focused on Physics education, creating engaging content to help students across all levels. Now an experienced GCSE and A Level Physics and Maths tutor, Ashika helps to grow and improve our Physics resources.
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