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 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 Fx= F cos 30 and Fy= 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.
Fy = 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.
FAQs on Components of a Vector
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:
\[ \overrightarrow{a} \] = ax.î + ay.ĵ + az.k̂
Here,
ax is called the magnitude of the x-component of the given vector \[ \overrightarrow{a} \]
ay is called the magnitude of the y-component of the given vector \[ \overrightarrow{a} \], and,
az is called the magnitude of the z-component of the given vector \[ \overrightarrow{a} \].
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 \[ \overrightarrow{a} \] 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 ay = a × sin θ
Where,
ax is called the magnitude of the x-component of the given vector \[ \overrightarrow{a} \].
ay is called the magnitude of the y-component of the given vector \[ \overrightarrow{a} \].
Resolving a Three-Dimensional Vector
Consider a to be the magnitude of the vector \[ \overrightarrow{a} \] , 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,
ay = a × sin θ, and,
az = √(aₓ + aᵧ × sinθ)
Where,
ax is called the magnitude of the x-component of the given vector \[ \overrightarrow{a} \]
ay is called the magnitude of the y-component of the given vector \[ \overrightarrow{a} \], and,
az is called the magnitude of the z-component of the given vector \[ \overrightarrow{a} \].
3. What will be the x and y components of a vector that has a magnitude of 12 and makes an angle of 45 degrees with the positive x-axis?
According to the given question, the vector has a magnitude of 12 and makes an angle of 45 degrees. So let V be 12 and θ be 45 degrees. In order to calculate the x component of the vector, calculate Vcosθ which will be the product of 12 and cos 45. Since the value of cos 45 is \[ \frac{1}{\sqrt{2}}\]. On putting the value of cos 45, the x component of the vector on calculation will come out to be \[ 6\sqrt{2}\]. Now, in order to calculate the y component of the vector, calculate vsinθ which will be 12sin45. The value of sin 45 is the same as cos 45, which is \[ \frac{1}{\sqrt{2}}\]. Therefore, on putting the value of sin 45, in the equation, the y component will be \[12 \times \frac{1}{\sqrt{2}}\], which will come out as \[ 6\sqrt{2}\] after calculation. Therefore, in the given question the x and the y component will be the same, that is, \[ 6\sqrt{2}\].
4. What are the applications of vectors and its components?
Vectors are a very useful way to represent physical quantities because they are a combination of both magnitude and direction, which is represented in the form of arrows. The vectors and their components are widely used as a graphical tool for representing the position, velocity, displacement, and acceleration. The position vector is used to analyse the object from the origin in a coordinate system. The position vectors are also used for analysing the position of the object with respect to a specific reference point, initial position, or secondary object.
The position vector basically refers to a line that is drawn from an imaginary origin to the object so that the quantity has both a direction and a magnitude making it a vector quantity. The components of a vector are also very beneficial for calculating the velocity of a moving object because even velocity is expressed in terms of direction and magnitude. Since the acceleration of a moving object is also expressed in terms of magnitude and direction, therefore, the acceleration also uses the components of the vector to give useful insights to the study.
5. What are the major terminologies used in the study of vectors and their components?
In order to understand the vectors and their components well, you must get familiar with terminologies like coordinates (numbers that indicate the position of an object with respect to a certain axis), axis (an imaginary line around which it is believed that the object spins or is arranged in a symmetric manner), magnitude (the number that is assigned to a vector to indicate its length), coordinate axis (perpendicular lines that can define the coordinates in relation to the origin), origin (refers to the centre of the coordinate axis), component, vector (directed quantity having both magnitude and direction), scalar (quantity having only magnitude and no direction), unit vector (vector having magnitude 1), velocity, acceleration, and displacement.
