Components of a Vector

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 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.

Vector and its Components - At A Glance

• Vector has two components in which it can be broken, that is, magnitude and direction.

• By using the hypotenuse method, we can calculate the horizontal component and vertical component of the vector by using the angle that the vector makes with the two components.

• Scalar quantities (example, mass, height, volume, and area) are physical quantities that are represented by a single number whereas vector quantities (example, velocity, displacement, and acceleration) are quantities that are represented in the form of two components, that is, direction and magnitude.

• Vector quantities can be broken down into components of the horizontal and vertical axis.

• The vector that has a magnitude of 1, is known as a unit vector.

• Vectors are majorly the arrows with a magnitude and direction, therefore if a vector represents any quantity, then that quantity has both magnitude and direction.

• The most common physical quantities which are represented in the form of vectors are displacement, acceleration, and velocity.

• Since acceleration represents the rate of change of velocity with respect to time, requires both direction and magnitude.

• Displacement, velocity, and acceleration are all related to each other because the calculation of one requires the value of the other due to which all three are vector quantities.

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,

\[ |v| = \sqrt{v_{x}^{2} + v_{y}^{2}}\]

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 Pythagorean Theorem, you get,

\[ |v| = \sqrt{v_{x}^{2} + v_{y}^{2}}\]

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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2. 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 F_x= F cos 30 and F_y= F sin 30. 

Solving them you get,

F_x= F cos 30 = 20 x cos 30

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

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

F_y = F sin 30 = 20 x sin 30

= 20 x 0.5

Hence your answer is 10 Newton.

What will be the Outcomes of Studying Vectors and their Components?

Vectors and their components are a very essential chapter taught to students and a lot of questions are asked from this chapter in both school exams as well as competitive exams. The advanced version of vectors is also included in the future chapters. The major outcome of learning vectors and their components will be that the students will be able to differentiate between firstly, two-dimensional and three-dimensional vectors and secondly, between scalar quantities and vector quantities. The students will be able to design a graphical model for calculations like addition and subtraction of vectors, and will also be able to summarise the relationship between scalar quantities and vector quantities. The students will further be able to interpret the influence of the product of a scalar and vector quantity and will be able to give value to the applications of vectors in the field of Physics.