explain the polygon law of vector addition?

Asked by aashu.60 | 22nd Jul, 2015, 11:37: AM

Expert Answer:

Polygon law of vector addition states that if a number of vectors can be represented in magnitude and direction by the sides of a polygon taken in the same order, then their resultant is represented in magnitude and direction by the closing side of the polygon taken in the opposite order.
Let us find the resultant of four vectors begin mathsize 14px style straight A with rightwards arrow on top comma space straight B with rightwards arrow on top comma space straight C with rightwards arrow on top space and space straight D with rightwards arrow on top end style.
In traingle OKL, the vectors begin mathsize 14px style straight A with rightwards arrow on top space a n d space straight B with rightwards arrow on top end style are represented by the sides begin mathsize 14px style OK with rightwards arrow on top space a n d space KL with rightwards arrow on top end style taken in the same order. Therefore, from the triangle law of vector addition, the closing side begin mathsize 14px style OL with rightwards arrow on top end style taken in the opposite order represents the resultant of vectors begin mathsize 14px style OR with rightwards arrow on top space a n d space KL with rightwards arrow on top end style.
Thus, begin mathsize 14px style OK with rightwards arrow on top space plus space KL with rightwards arrow on top space equals space OL with rightwards arrow on top space end style.     ... (1)
By applying the traingle law of vector addition to the traingle OLM, It shows that the side begin mathsize 14px style OM with rightwards arrow on top end style is the resultant of vectors begin mathsize 14px style stack O straight L with rightwards arrow on top space a n d space LM with rightwards arrow on top space end style i.e., begin mathsize 14px style OL with rightwards arrow on top space plus space LM with rightwards arrow on top space equals space OM with rightwards arrow on top space end style.
Using eg (1), we get,
begin mathsize 14px style OK with rightwards arrow on top space plus space KL with rightwards arrow on top space plus space LM with rightwards arrow on top space equals space OM with rightwards arrow on top space end style    .... (2)
Similarly applying the triangle law of vector addition to the triangle OMN, we get,
begin mathsize 14px style OM with rightwards arrow on top space plus space MN with rightwards arrow on top space equals space ON with rightwards arrow on top space end style
Using eq (2), we get,
begin mathsize 14px style OK with rightwards arrow on top space plus space KL with rightwards arrow on top space plus space LM with rightwards arrow on top space plus space MN with rightwards arrow on top equals space ON with rightwards arrow on top space end style      ... (3)
Now the vectors begin mathsize 14px style OK with rightwards arrow on top space equals straight A with rightwards arrow on top comma space space KL with rightwards arrow on top space equals space straight B with rightwards arrow on top comma space LM with rightwards arrow on top space equals space straight C with rightwards arrow on top space and space MN with rightwards arrow on top equals straight D with rightwards arrow on top end style.
Denoting the vector, begin mathsize 14px style ON with rightwards arrow on top equals straight R with rightwards arrow on top end style, the equation becomes,
begin mathsize 14px style straight A with rightwards arrow on top plus straight B with rightwards arrow on top plus straight C with rightwards arrow on top plus straight D with rightwards arrow on top equals straight R with rightwards arrow on top end style

Answered by Faiza Lambe | 22nd Jul, 2015, 12:22: PM