Components of a Vector

What are Vector Components?

Any vector that is directed in two dimensions can be thought to be having an influence in two different directions. This means that it can be thought to have two different parts. Each part of the two-dimensional vector is called a component. The components of a vector helps to depict the influence of that vector in a particular direction. The combined influence of both these components is equal to the influence of the two-dimensional single vectors. The single two-dimensional vector can be replaced by the two vector components.


The components of a vector in the two-dimension coordinate system are generally considered to be the x-component and the y-component. You can represent it as, 

V = \[(v_{x}, v_{y})\]


where V is called as the vector. 


These are the parts of the vectors that are generated along the axes of the coordinate system. In this article, you would be finding the components of a given vector by using the formula for both the two-dimensional and the three-dimensional coordinate system.


Components of a Vector Definition

In the two-dimensional coordinate system, you can break down any vector into its x-component and y-component.  This is denoted as:


\[\overrightarrow{v}\] = \[(v_{x}, v_{y})\]


Consider the following example:


In the diagram shown below, the vector v is divided into two of its components that are \[v_{x}\] and .\[v_{y}\].


Consider the angle between the vector and its x -component to be θ.

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The vector and the vector components here form a right angle triangle as shown below:

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The trigonometric ratios would give you the relation between the vector magnitude and the vector components. 


\[cos\Theta = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{v_{x}}{v}\]


\[sin\Theta = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{v_{y}}{v}\]


\[v_{x} = vcos\Theta\]


\[v_{y} = vsin\Theta\]


When you use the Pythagoras Theorem in the right-angled triangle with lengths \[v_{x}\] and \[v_{y}\], you get,


|v| = \[\sqrt{v^{x2} + v^{y2}}\]


Components of a Vector Formula

As you know,


\[cos\Theta = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{v_{x}}{v}\]


\[sin\Theta = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{v_{y}}{v}\]


The components of a vector formula is derived as


\[v_{x} = vcos\Theta\]


\[v_{y} = vsin\Theta\]


Using the Pythagoras Theorem, you get,


|v| = \[\sqrt{v^{x2} + v^{y2}}\]


  1. Components of a Two - Dimensional Vector

Consider for example a two-dimensional vector \[\overrightarrow{a}\] that has an initial point O in the coordinate system and has a final point A as well.


Now, if you produce lines from the points O and A such that they meet at a point C and make a 90angle with one another, you would get two newly formed vectors \[\overrightarrow{a_{x}}\] and \[\overrightarrow{a_{y}}\].


These are said to be the components of the vector \[\overrightarrow{a}\].

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  1. Components of a Three - Dimensional Vector

Just like the two-dimensional components of a vector if you resolve the given vector \[\overrightarrow{a}\]into its components in the three-dimensional system having the x, y, z axes, you get,

\[\overrightarrow{a_{x}}\], \[\overrightarrow{a_{y}}\] and \[\overrightarrow{a_{z}}\]


The three newly formed vectors are known as x, y, z components of a vector in 3D respectively of the vector \[\overrightarrow{a}\].


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Components of Vector Example

The magnitude of a given vector F and the direction of its vector is 60along the horizontal. Find its vector components.

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Solution


\[F_{x} = F cos60\]


Solving this gives you \[10\times \frac{1}{2} = 5\].


\[F_{y} = F sin60\]


Solving this gives you \[10\times \frac{\sqrt{3}}{2} = 5\sqrt{3}\]


Hence, the vector F is equal to 5, \[5\sqrt{3}\].


Vector Components Problem

A force of 20 N makes an angle of 30 degrees with the x-axis. Find both the x-component and the y component of the given force.


Solution


The first step is to draw the diagram. Your diagram would look like this:

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Then find out the vector components of the given force of 20 N.

To do so, find out Fₓ= F cos 30 and Fᵧ= F sin 30. 

Solving them you get,


Fₓ= F cos 30 = 20 x cos 30


= \[(20)(0.5\sqrt{3})\]


Hence, your answer is \[10\sqrt{3}\] Newton.


Fᵧ = F sin 30 = 20 x sin 30


= 20 x 0.5


Hence your answer is 10 Newton.

FAQ (Frequently Asked Questions)

1. What is the Notation of a Unit Vector?

You can denote a vector by using the concept of components of a vector and unit vector and write them in the unit vector form as shown:


a = aₓ.î + aᵧ.ĵ + az.k̂


Here,


aₓ is called the magnitude of the x-component of the given vector a

aᵧ is called the magnitude of the y-component of the given vector a, and,

az is called the magnitude of the z-component of the given vector a.

2. How to Resolve a Vector into its Components?

For resolving a vector into its components, you can use the following formulas:


Resolving a two-dimensional vector into its components


Consider a to be the magnitude of the vector a and θ to be the angle that is formed by the vector along the x-axis or to be the direction of the given vector. Then, you would get,


aₓ = a × cos θ and aᵧ = a × sin θ


Where,

aₓ is called the magnitude of the x-component of the given vector a.

aᵧ is called the magnitude of the y-component of the given vector a.


Resolving a three-dimensional vector


Consider a to be the magnitude of the vector a, θ to be the angle that is formed by the vector along the x - axis and ዋ to be the angle formed by the vector along the x-y plane.


Then you get,


aₓ = a × cos θ,


aᵧ = a × sin θ, and,


az = √(aₓ + aᵧ × sinዋ)


Where,

aₓ is called the magnitude of the x-component of the given vector a

aᵧ is called the magnitude of the y-component of the given vector a, and,

az is called the magnitude of the z-component of the given vector a.