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

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

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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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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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, \[5\sqrt{3}\].

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 + a_{z}.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,

a_{z} 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,

a_{z} = √(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,

a_{z} is called the magnitude of the z-component of the given vector a^{→}.