# 4.1.2 Resolving Vectors

### Resolving Vectors

• Two vectors can be represented by a single resultant vector
• Resolving a vector is the opposite of adding vectors
• A single resultant vector can be resolved
• This means it can be represented by two vectors, which in combination have the same effect as the original one
• When a single resultant vector is broken down into its parts, those parts are called components
• For example, a force vector of magnitude F and an angle of θ to the horizontal is shown below

The resultant force F at an angle θ to the horizontal

• It is possible to resolve this vector into its horizontal and vertical components using trigonometry

The resultant force F can be split into its horizontal and vertical components

• For the horizontal component, Fx = F cos θ
• For the vertical component, Fy = F sin θ

#### Forces on an Inclined Plane

• Objects on an inclined plane is a common scenario in which vectors need to be resolved
• An inclined plane, or a slope, is a flat surface tilted at an angle, θ
• Instead of thinking of the component of the forces as horizontal and vertical, it is easier to think of them as parallel or perpendicular to the slope
• The weight of the object is vertically downwards and the normal (or reaction) force, R is always vertically up from the object
• The weight W is a vector and can be split into the following components:
• W cos (θ) perpendicular to the slope
• W sin (θ) parallel to the slope
• If there is no friction, the force W sin (θ) causes the object to move down the slope
• The object is not moving perpendicular to the slope, therefore, the normal force R = W cos (θ)

The weight vector of an object on an inclined plane can be split into its components parallel and perpendicular to the slope

#### Worked Example

A helicopter provides a lift of 250 kN when the blades are tilted at 15º from the vertical.

Calculate the horizontal and vertical components of the lift force.

Step 1: Draw a vector triangle of the resolved forces

Step 2: Calculate the vertical component of the lift force

Vertical = 250 × cos(15) = 242 kN

Step 3: Calculate the horizontal component of the lift force

Horizontal = 250 × sin(15) = 64.7 kN

#### Exam Tip

If you’re unsure as to which component of the force is cos θ or sin θ, just remember that the cos θ is always the adjacent side of the right-angled triangle AKA, making a ‘cos sandwich’

### Equilibrium

• Coplanar forces can be represented by 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:

Three forces on an object in equilibrium form a closed vector triangle

#### 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 could ask you to draw 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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