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
Example: 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
Examiner 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'