Resolution of Vectors

1. Horizontal component (along the x-axis)
2. Vertical component (along the y-axis)

Suppose a force $F$ acts at an angle $\theta$ from the horizontal or vertical:

1. If $\theta$ is with respect to the horizontal, then:

Horizontal component:

$F_x = F \cos \theta$

Vertical component:

$F_y = F \sin \theta$

2. If $\theta$ is with respect to the vertical, then:

Horizontal component:

$F_x = F \sin \theta$

Vertical component:

$F_y = F \cos \theta$

Problem: Find the horizontal and vertical components of a force of 10 N acting at 30° to the vertical.

Since angle is with respect to the vertical:

Horizontal component

$F_x = F \sin \theta$

Vertical component

$F_y = F \cos \theta$

Solution:

Given:

$F = 10\, \text{N},\ \theta = 30^\circ$

Step 1: Horizontal component

$F_x = F \sin \theta = 10 \sin 30^\circ$

$F_x = 10 \times 0.5 = 5\, \text{N}$

Step 2: Vertical component

$F_y = F \cos \theta = 10 \cos 30^\circ$

$F_y = 10 \times 0.866 = 8.66\, \text{N}$

Answer:

• Horizontal Component = 5 N
• Vertical Component = 8.66 N