# Components Of Vectors

## Vectors in physics. Vectors and its components. Splitting of vectors into its components. Found of the components of vectors. Addition of the components of the vectors.

**1- Splitting A Vector Into Its Components**

To get the vertical and horizontal components of the vectors, the vector are separates into two components that it’s perpendicular to each other using coordinate system.

If the angle between the x axis with the vector is "α", the components of the vector is found as the following.

Cosα = | F_{x} | |

F |

F_{x} = F•cosα

Sinα = | Fy | |

F |

F_{y} = F•sinα

F_{x}: The horizontal component.

F_{y}: The vertical component.

**Example:**

The magnitude of the vector F_{1} that seen in the figure is 80 N, the angle between with x axis and its is 53 degrees.

The magnitude of the vector F2 is 50 N, and the angle between with x axis and its is 53 degrees.

What is the resultant of the F_{1} + F_{2}

(sin53° = 0.8, cos53°=0.6, cos37°=0.8, sin37°=0.6)

**Solution:**

We separates the vectors into its components and add the components.

First, let we separate the vectors into its components.

The components of the vector F_{1}.

Cosα = | F_{x} | |

F_{1} |

F_{1x} = F_{1}•cos53°

F_{1x} = 80•cos53°

F_{1x} = 80•0.6

F_{1x} = 48 N

Sin53 = | F_{1y} | |

F_{1} |

F_{1y} = F1•sin53°

F_{1y} = 80•sin53°

F_{1y} = 80•0.8

F_{1y} = 64 N

**The components of the vector F2.**

The vector F2 is doing angle of 53 degrees from left side with x axis. The angle that vector F2 doing with x axis is 180 + 53 = 233 degrees.

Cos233° = Cos(180 + 53) = -Cos53°

F_{2x} = - 50•0.6

F_{2x} = - 30 N

Sin233° = Sin(180 + 53) = -Sin53°

F_{2y} =- 50•0.8

F_{2y} = -40 N

Now, let we add, the components with the same direction.

F_{tx} = F_{1x} + F_{2x}

F_{tx }= 48 – 30

F_{tx} = 18 N

F_{ty} = F_{1y} + F_{2y}

F_{ty} = 64 - 40

F_{ty} = 24 N

Now, we have two vectors that are perpendicular to each other.

The resultant vector can be found using pythagorean theorem.

F_{t}^{2} = 18^{2} + 24^{2}

(F_{t})^{2 } = 324 + 576

(F_{t})^{2} = 900

F_{t} = 30 N

If we want find the angle that vector Ft does with x axis, we can use equality in the following.

tanα = | 24 | |

18 |

tanα = 1.33

α = tan^{-1 }1.33

α = 53 degrees.

**Example:**

Find the resultant of three forces that affected to the object K.

(sin53° = 0.8, cos53°=0.6, cos37°=0.8, sin37°=0.6)

**Solution:**

First we must separate the forces into its components.

Next, we add the components that have same direction among themselves.

So, we add the components that on the x axis among themselves and we add the components that on the y axis among themselves too.

F_{1x} = 200•cos( – 37)

F_{1x} = 200•0.8

F_{1x} = 160 N

F_{1y} = 200•sin(- 37)

F_{1y} = -200•0.6

F_{1y} = - 120 N

F_{2x} = 50•cos37°

F_{2x} = 50•0.8

F_{2x} = 40 N

F_{2y} = 50•sin37°

F_{2y} = 50•0.6

F_{2y} = 30 N

F_{3x} = 100•cos(-37)

F_{3x} = 100•0.8

F_{3x} = 80 N

F_{3y} = 100•sin(-37)

F_{3y} = -100•0.6

F_{3y} = -60 N

Let we add the components among themselves.

F_{x} = F_{1x} + F_{2x} + F_{3x}

F_{x} = 160 + 40 + 80

F_{x} = 280 N

F_{y} = F_{1y} + F_{2y} + F_{3y}

F_{2} = -120 + 30 – 60

F_{2 }= - 150 N

Finally, we have two forces perpendicular to each other. We can find the resultant of these vectors using pythagorean theorem.

F = 150^{2} + 280^{2}

F = 22500 + 78400

F = 100900

F = 317.65 N

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14/09/2018

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