Resolving a vector splits it into two perpendicular components

Scalars & Vectors — AQA A-Level Physics

Resolving is the reverse of adding vectors.
A single vector can be split into two components that, combined, have the same effect as the original.
For a vector F at angle theta to the horizontal: the horizontal component is F cos(theta) and the vertical component is F sin(theta).

F_x = F \cos \theta

$F_x$: horizontal component
$F$: magnitude of the vector
$θ$: angle between the vector and the horizontal

F_y = F \sin \theta

$F_y$: vertical component
$F$: magnitude of the vector
$θ$: angle between the vector and the horizontal

Worked Example

A helicopter provides a lift of 250 kN when the blades are tilted at 15 degrees from the vertical. Calculate the horizontal and vertical components of the lift force.

Show Solution

Identify the angle

The force is 15 degrees from the vertical, so the vertical component uses cos and the horizontal uses sin.

Calculate the vertical component

$$F_V = 250 \times \cos 15° = 242 \text{ kN}$$

Calculate the horizontal component

$$F_H = 250 \times \sin 15° = 64.7 \text{ kN}$$

Answer

$Vertical = 242 kN, Horizontal = 64.7 kN.$

Common Mistake MEDIUM

Students often: Swapping sin and cos when resolving vectors.
Instead: The cos component is always the one adjacent to the angle (the 'cos sandwich'). If the angle is from the horizontal, the horizontal component is F cos(theta). If the angle is from the vertical, the vertical component is F cos(theta).

Related:Kinematics