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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:

Then we draw the rectangle with F_{x} and F_{y} as two of the sides:

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

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

F = √F_{x}^{2} + F_{y}^{2}

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 F_{x}:

According to trigonometry:

θ = tan^{-1} | F_{y} |

F_{x} |

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:

F_{x} > 0

F_{y} = 0

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

F_{y} 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 F_{x}:

F = F_{x}

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

F_{x} < 0

F_{y} = 0

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

And since F_{x} is negative, the magnitude will be −F_{x} (remember a magnitude is *always positive*), therefore:

F = −F_{x}

So if F_{x} 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:

F_{x} = 0

F_{y} = 0

↓

F = 0

To test your understanding, make sure to do the exercises below.

## Exercises

### #1

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

### Solution

F_{x} = −7.0 N

F_{y} = 0 N

F will be in the negative x direction, and have the same magnitude as the x component:

F = −F_{x} = 7.0 N

It is −F_{x} because F_{x} is negative, and the magnitude must be positive.

### #2

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

### Solution

We first roughly represent F_{x} and F_{y} on an xy-plane, and from that we draw the rectangle and F.

F_{x} = 50.0 N

F_{y} = 21.2 N

Let's find the magnitude of F applying **Pythagoras' Theorem:**

F = √F_{x}^{2} + F_{y}^{2}

F = √50.0^{2} + 21.2^{2} N

F = √2949 N

F = 54.3 N

Next we find θ:

θ = tan^{-1} | F_{y} |

F_{x} |

θ = tan^{-1} | 21.2 N |

50.0 N |

θ = tan^{-1} 0.424

θ = 23.0°

In this case θ is already the direction angle of F. Indeed θ is the counterclockwise angle that F makes with the positive x axis.

Therefore the force has magnitude 54.3N and the direction angle is 23.0°.

### #3

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

### Solution

F_{x} = −387 N

F_{y} = −532 N

Let's determine the magnitude of F:

F = √F_{x}^{2} + F_{y}^{2}

F = √(−387)^{2} + (−532)^{2} N

F = 658 N

And then θ:

θ = tan^{-1} | F_{y} |

F_{x} |

θ = tan^{-1} | −532 N |

−387 N |

θ = tan^{-1} 1.37

θ = 53.9°

We need the direction angle of F, i.e. the counterclockwise angle F makes with the positive x axis.

Looking at the xy-plane above, we see that we just need to add 180° to θ. Therefore the direction angle of the force will be 53.9°+180°=233.9°:

Hence, the magnitude is 658N and the direction angle is 233.9°.

### #4

Find F knowing that F_{x} is −9.48N and F_{y} 5.67N.

### Solution

F_{x} = −9.48 N

F_{y} = 5.67 N

The magnitude of F will be:

F = √F_{x}^{2} + F_{y}^{2}

F = √(−9.48)^{2} + (5.67)^{2} N

F = √122 N

F = 11.0 N

And θ:

θ = tan^{-1} | F_{y} |

F_{x} |

θ = tan^{-1} | 5.67 N |

−9.48 N |

θ = tan^{-1} (−0.598)

θ = −30.9°

Since the tangent is negative, θ came out negative. But we are just interested in the magnitude, so we ignore the minus sign:

θ = 30.9°

The direction angle of F will be 180°−θ (look at the figure above), i.e. 180°−30.9°=149.1°:

### #5

F has the following components: 0N in the x direction, and 8.3×10^{2}N in the y direction. Determine magnitude and direction of F.

### Solution

F_{x} = 0 N

F_{y} = 8.3 × 10^{2} N

F will be in the direction of the positive y axis, and have the same magnitude as F_{y}:

F = F_{y} = 8.3 × 10^{2} N

### #6

F_{x} is 0.41N, F_{y} is −0.80N. Find F.

### Solution

F_{x} = 0.41 N

F_{y} = −0.80 N

Let's first determine the magnitude of F:

F = √F_{x}^{2} + F_{y}^{2}

F = √(0.41)^{2} + (−0.80)^{2} N

F = √0.808 N

F = 0.90 N

Next let's find θ:

θ = tan^{-1} | F_{y} |

F_{x} |

θ = tan^{-1} | −0.80 N |

0.41 N |

θ = tan^{-1} (−1.95)

θ = −63°

θ is negative because the tangent is negative. But we are just interested in the magnitude of the angle, so we can ignore the minus sign:

θ = 63°

Looking carefully at the xy-plane above, we can see that the direction angle of F is θ subtracted from 360°, i.e. 360°−63°=297°:

## You may also want to read:

- How to decompose a force into x and y components
- Solving problems which involve forces, friction, and Newton's Laws: A step-by-step guide
- What is the Resultant Force and How to Find it (with Examples)
- What is a Free-Body Diagram and How to Draw it (with Examples)