Graphical representation of Vectors

1 (b) Determine the resultant of two vectors by a graphical method.

In the previous post, we learned that vectors are quantities which have a magnitude and a direction. As vector quantities have a direction too, we can represent them graphically. Graphical representation of vectors is extremely easy. All it takes is some practice and understanding, and you will be on your way to master this topic.

In graphical representation of vectors, what we do is represent vectors on a sheet of paper, in two dimensions. Usually in exams, the questions come about the resultant of two vectors. The first thing we need to learn about is the addition of vectors.



Head to Tail Method for addition of Vectors

The head to tail method is a simple way to add vectors. Lets say we have to vectors, A and B. 




 Now we need to draw them such that the tail of vector B is in contact with the head of vector A like this.




 The resultant vector, let's name it R, is found out by joining the tale of the first vector with the head of the second vector. Note that the tale of vector R is at the tale of vector A, the first Vector. And the head of vector R is at the head of vector B, the second vector.



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Now that you understand the method for addition of vectors, I'll come back to the main topic. To graphically represent vectors, the first thing we need to do is to take a suitable scale. By scale, i mean the ratio of the vectors and the lengths that we are going to use to represent those vectors on a paper e-g 1 cm to represent 1000 N. The next step is to draw the two vectors on a piece of paper according to the scale and the direction given. The resultant vector is then found out by the head to tail method of addition of vectors.




To help you better understand the topic, I'll be solving a simple question for you, and fill in the details step by step.

Question: A man walks 1200 m towards the North, and then 600 m towards the East. Find the resultant displacement of the man by graphical representation.

Solution.




Step 1: The first thing we need to do is to decide a suitable scale, lets say 1 cm to represent 100m on our paper.

Step 2:  Now we have to draw the diagram on a piece of paper, according to the scale. Our first displacement is towards the North, of 1200 m. Now according to our scale, as 1 cm represents 100m, I'll divide 1200 by 100. The answer would be 12. So our first displacement would be of 12cm towards the North. Similarly, i found out that our second displacement would be of 6 cm towards the East. I'll now draw the two vectors, and join the head of the first vector with the tale of the second vector. (Note, this diagram is not drawn to scale)



Step 3: Now draw the resultant vector according to the vector addition method.



Step 4: In this question, the length of the resultant vector would be approximately 13.42 cm. To find the resultant displacement, I'll now multiply 13.42 with 100, according to our scale. The resultant displacement would be of 1342m. Question Solved!




But sometimes, the questions are a bit more difficult. The two vectors are not so easy because they are not perpendicular to each other. I am solving once such question for you guidance.


Question: A force of 2000N acts on an object in the North-East direction and another force of 1200N acts on the same object in the South East Direction. The angle between the two vectors is 155 degrees. Find the resultant of the two forces.

Solution: Now this question is a bit tricky. But as before, the first thing to do is to choose an appropriate scale. I am choosing a scale of 1 cm to represent 100N. (Note that the following diagram is not to scale)


The head to tail method is applied here too, but in a bit different way. You will need to get your hands on a compass. Open your compass 20 cm, put it on point C and make an arc. Now open your compass 12 cm, put it on B, and make another arc so that it cuts the first arc. Mark this point D. Now join B and C with D, to complete the parallelogram.



Now join A with D, this is your resultant force. Multiple the length of this vector with 100, the scale, and you will have your resultant force.



1 comment:

  1. Hey.. can i get all these note in one file . i mean as a document or pdf. it will be then very helpful

    ReplyDelete