Download Components of vectors

Components of vectors
It is often necessary to find the components of a vector, usually in two perpendicular directions.
This process is called the resolution of a vector. What you are really doing is finding the
effectiveness of the vector along a specified direction.
The component of a vector along any direction is the magnitude of the vector multiplied by the
cosine of the angle between the vector and the line.
F
F sin A
F
Figure 1
A
Figure 2
A
F cosA
F cos A
The horizontal component of the vector F shown in Figure 1 is F cos (A) while Figure 2 shows the
components of a vector in two perpendicular directions. These are known as the rectangular
components of the vector.
Component Fx = F cos A
Component Fy = Fcos (90 - A) = F sin A
sin A = O/H
hypotenuse (H)
opposite (O)
A
cos A = A/H
tan A = O/A
adjacent (A)
Imagine pulling a barge along a canal or a truck along a track by a rope inclined at an angle to the
track (Figure3). The smaller the angle the more effective the force in the rope (the cosine of the
angle gets bigger when the angle gets smaller).
F sin A
F
A
F cos A
Figure 3
1
The purpose of the following diagrams is to see if you can work out the components of each vector
although the diagrams are twisted round at odd angles.
Figure 4
A
The force in the towrope between a tug and a large liner has both a vertical and a horizontal
component.
Force in towrope has both a vertical and a
horizontal component
Figure 5
A projectile
Resolution of vectors is especially useful
when considering problems like the motion of
a projectile (Figure 6). Its velocity at any point
on its path is the combination of a horizontal
component (vx) (constant if there is no air
resistance) and a vertical component (vy)
which varies as time goes by. This vertical
component is maximum at the bottom of the
path and zero at the top.
If A is the angle of projection (relative to the
horizontal) and the velocity of projection is u
then:
Horizontal component = vx = u cos A
Initial vertical component = vy = u sin A
In the diagram the blue vectors represent the
actual velocity at any point while the red
vectors represent the horizontal and vertical
components of these velocities.
vx
Vy
vx
vx
Vy
Vy
vx
vx
Vy
Vy
Figure 6
vx
Vy
2