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How to find the magnitude and direction of a force given the x and y components

Sometimes we have the x and y components of a force, and we want to find the magnitude and direction of the force.


Let's see how we can do this.


There are three possible cases to consider:


• The two components are both different from zero

If a force F has the x and y components both different from zero, in order to find F we start by roughly representing the components on an xy-plane (based on the magnitude of the components and their sign).


If, for instance, both the components are positive, with the x component slightly larger in magnitude, we would represent them something like this:

The x and y components of a force, both different from zero

Then we draw the rectangle with Fx and Fy as two of the sides:

The rectangle that has Fx and Fy as two of its sides

The diagonal of the rectangle that goes from the origin is the force F:

The force F is the diagonal of the rectangle

We can find the magnitude of F, by applying Pythagoras' Theorem:

F = Fx2 + Fy2

And what about the direction of F ?


The direction is often expressed by the direction angle, i.e. the counterclockwise angle that F makes with the positive x axis.


Let's see how we can find it:


First we find θ, the angle F makes with its component Fx:

Angle theta formed by F and its x component

According to trigonometry:

θ = tan-1 Fy
Fx

After we have found θ, we can easily determine the direction angle.


Sometimes θ will already be the direction angle, other times you will need to add θ to 180° or subtract it from 180° etc., it depends in what quadrant your force is.

Check out the exercises below to see some examples.


• One of the two components is equal to zero

Often a force has either the x or y component equal to zero and the other component different from zero.

In that case, the magnitude and direction of the force is equal to the magnitude and direction of the non-zero component:


For example let's assume that a force F has y component zero, and x component > 0:

Fx > 0
Fy = 0

If we represent the two components graphically, we should see something like this:

The x component of the force F is greater than 0

Fy is zero, so we can't actually see it.


It is clear that F will be in the direction of the positive x axis and have the same magnitude as Fx:

The force F is in the direction of the positive x axis
F = Fx

On the other hand, if the x component of F is negative,

Fx < 0
Fy = 0
The x component of the force F is lesser than 0

F will be in the negative direction of the x axis, and the magnitude will be the same as that of Fx.

And since Fx is negative, the magnitude will be −Fx (remember a magnitude is always positive), therefore:

The force F is in the direction of the negative x axis
F = −Fx

So if Fx is −10N, then F has magnitude 10N.



The same can be shown for a force that has the x component equal to zero, and the y component different from zero.


• The two components are both equal to zero

If both the components are equal to zero, then the force is also equal to zero:

Fx = 0; Fy = 0
F = 0

To test your understanding, make sure to do the exercises below. Also, if you want to see this technique in action, check the step-by-step guide for solving force problems.


Exercises


#1

The x component of a force is −7.0N, the y component is 0N. Find the magnitude and direction of the force.


#2

Find a force knowing that its x and y components are 50.0N and 21.2N respectively.


#3

Assuming that a force has the x component −387N and the y component −532N, find magnitude and direction of the force.


#4

Find F knowing that Fx is −9.48N and Fy 5.67N.


#5

F has the following components: 0N in the x direction, and 8.3×102N in the y direction. Determine magnitude and direction of F.


#6

Fx is 0.41N, Fy is −0.80N. Find F.


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