Resolution of vector into component, Addition and subtraction of vector in coordinate form, Practice problem, FAQs

Suppose you are playing a game in which your eyes are covered with black cloth and you are searching for your friend.Lets your friend is 1 meter away in the east direction. Now by mistake you start moving somewhere between north and east and after moving 1 meter you find that He is not there. Do you know how much distance you have covered in the east direction? You can find it by resolving the vector which is denoting your position. Let's learn how to resolve a vector.

Table of content

Resolution of vector into component
Vector addition and subtraction using component
Practice problem
FAQs

Resolution of vector into component

Consider a rod which is placed in the x-y plane. If you see the rod from the top along the z-axis you will see the original length of it. Now if you see this from the side (says along x-axis) you will see that the length of the rod appears reduced, which will be equal to the projection of length on y-axis.This is called the component of length along y-axis. Similarly as seen along the y-axis the length will be lesser from the original and will be equal to the projection of length on the x-axis. This is called the component of length along the y-axis.

Any physical quantity that is given by a vector can be represented as the sum of two or more scaled unit vectors. The process of obtaining a scaled unit vector is called resolution of vectors. And the vectors formed after resolution are called components vectors.

Consider a 3 dimensional coordinate system in which a vector is showing. It can be resolved along , y and z direction as , and respectively.

Now, vector can be written as,

Here is the x-component of , is the y-component of , is the z-component of

And , and are the unit vector in , y and z direction respectively.

Let make an angle with x-axis, with y-axis and with z-axis, then the magnitude of component is given by trigonometry as

For 2 dimensional coordinate system vector is shown in figure,

vector can be written as,

Let make an angle with x axis and and are the component of along x-axis and y-axis respectively, by using trigonometry

Then, vector

Where A is the magnitude of vector and is given by

And is the angle of vector with the x-axis

Vector addition and subtraction using component

We know how to add and subtract the vector using the geometric method. As the coordinate system is decided, the addition and substitution of vectors become easier.

Consider the two vectors and in the cartesian coordinate system can be expressed as

Then the addition of two vectors is given by simply adding their corresponding x, y and z components.

Similarly, the subtraction of two vectors is given by simply subtracting their corresponding x, y and z components.

These are the analytical ways of adding and subtracting the two vectors.

Practice problem

Q 1. Calculate the rectangular components of a force of that acts in a direction of North East?

Since there is in a right angle triangle we can deduce that the direction of force can be shown as East degrees North.

The force and its rectangular components are shown. One component , , is acting to the East , the other , , is to the North.

The magnitudes of these forces can be calculated:

Thus, the 10 Newton force can be resolved into two rectangular components: to the East and to the North.

Q 2. Two vector and are given as and . Find , .

Vector is equal to vector .

Q 3. Two vector and are given as and . Find , .

Vector is opposite to vector .

Q 4. Find the unit vector in the direction of the sum of the vectors and

Sum of and ,

Unit vector is,

FAQs

Q 1. Why are vectors resolved into components?
A. The components of a vector describes the influence of that vector in a different direction.

Q 2. What can be the maximum magnitude of a component of a vector?
A. Maximum magnitude of a component can be equal to the magnitude of the vector itself. In this case the vector will be parallel to the direction of the component.

Q 3. What are the two methods of vector resolution?

A.There are two method of resolution-

1. Graphical method
2. Trigonometric method

Q 4. What is the resultant vector of components of a vector?
A. The Resultant of the component of a vector is that vector itself of which the component is given.

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