CBSE Class 11-science Answered

1.State paralogram law of vector addition and derive magnitude and the direction of a resultant vector

Asked by preethiprithivi05 | 21 Feb, 2023, 09:28: PM

Expert Answer

Law of paralellogram of addition of two vectors states that if two vectors are represented as adjacent sides of parallelogram ,

then the resultant of these two vector is diagonal of parallelogram by manitude and direction.

Fig(1) shows two given vectors

begin mathsize 14px style a with rightwards arrow on top end style

and

begin mathsize 14px style b with rightwards arrow on top end style

. In Fig.(2), it is shown that the given vectors are adjanct sides OA and OB of parallelogram OACB.

Then the resultant vector

begin mathsize 14px style a with rightwards arrow on top space plus space b with rightwards arrow on top end style

is diagonal OC as shown in figure. (2)

If θ is angle between vectors

begin mathsize 14px style a with rightwards arrow on top end style

and

begin mathsize 14px style b with rightwards arrow on top end style

, then magnitude of resultant vector

begin mathsize 14px style a with rightwards arrow on top space plus space b with rightwards arrow on top end style

is determined by using law of cosines as shown in fig.(3)

In fig.(3) , ΔOBC is part of parallelogram shown in fig.(2) . Magnitude of side OB is magnitude of

begin mathsize 14px style b with rightwards arrow on top end style

and magnitude of BC is magnitude of

begin mathsize 14px style a with rightwards arrow on top end style

begin mathsize 14px style angle end style

OBC = ( 180 - θ ) , where θ is angle between vectors .

Magnitude of side OC equals to magnitude of resultant vector

begin mathsize 14px style a with rightwards arrow on top space plus space b with rightwards arrow on top end style

Magnitude of OC is determined from law of cosines as

begin mathsize 14px style open vertical bar a with rightwards arrow on top space plus space b with rightwards arrow on top close vertical bar squared space equals space open vertical bar a with rightwards arrow on top close vertical bar squared space plus space open vertical bar b with rightwards arrow on top close vertical bar squared space minus space 2 space open vertical bar a with rightwards arrow on top close vertical bar space space open vertical bar b with rightwards arrow on top close vertical bar space cos left parenthesis 180 minus theta right parenthesis end style

begin mathsize 14px style open vertical bar a with rightwards arrow on top space plus space b with rightwards arrow on top close vertical bar squared space equals space open vertical bar a with rightwards arrow on top close vertical bar squared space plus space open vertical bar b with rightwards arrow on top close vertical bar squared space plus space 2 space open vertical bar a with rightwards arrow on top close vertical bar space space open vertical bar b with rightwards arrow on top close vertical bar space cos theta end style

begin mathsize 14px style open vertical bar a with rightwards arrow on top space plus space b with rightwards arrow on top close vertical bar to the power of blank space equals space square root of stretchy vertical line a with rightwards arrow on top stretchy vertical line squared space plus space stretchy vertical line b with rightwards arrow on top stretchy vertical line squared space plus space 2 space stretchy vertical line a with rightwards arrow on top stretchy vertical line space space stretchy vertical line b with rightwards arrow on top stretchy vertical line space cos theta end root end style

Answered by Thiyagarajan K | 22 Feb, 2023, 12:12: PM

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