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, 21:28: PM
![](https://images.topperlearning.com/topper/tinymce/imagemanager/files/e0d223f6c9e88b1177ffdb1055d9532063f5b3a279e354.21862156f1.png)
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
and
. In Fig.(2), it is shown that the given vectors are adjanct sides OA and OB of parallelogram OACB.
![begin mathsize 14px style a with rightwards arrow on top end style](https://images.topperlearning.com/topper/tinymce/cache/d9ede463a364dc4ae9a804b43bf4b4bf.png)
![begin mathsize 14px style b with rightwards arrow on top end style](https://images.topperlearning.com/topper/tinymce/cache/1ee88d50cc6a096c65659dea54046215.png)
Then the resultant vector
is diagonal OC as shown in figure. (2)
![begin mathsize 14px style a with rightwards arrow on top space plus space b with rightwards arrow on top end style](https://images.topperlearning.com/topper/tinymce/cache/7cc00ac3f45ebe2a2f747b3eac2a3def.png)
If θ is angle between vectors
and
, then magnitude of resultant vector
is determined by using law of cosines as shown in fig.(3)
![begin mathsize 14px style a with rightwards arrow on top end style](https://images.topperlearning.com/topper/tinymce/cache/d9ede463a364dc4ae9a804b43bf4b4bf.png)
![begin mathsize 14px style b with rightwards arrow on top end style](https://images.topperlearning.com/topper/tinymce/cache/1ee88d50cc6a096c65659dea54046215.png)
![begin mathsize 14px style a with rightwards arrow on top space plus space b with rightwards arrow on top end style](https://images.topperlearning.com/topper/tinymce/cache/7cc00ac3f45ebe2a2f747b3eac2a3def.png)
In fig.(3) , ΔOBC is part of parallelogram shown in fig.(2) . Magnitude of side OB is magnitude of
and magnitude of BC is magnitude of ![begin mathsize 14px style a with rightwards arrow on top end style](https://images.topperlearning.com/topper/tinymce/cache/d9ede463a364dc4ae9a804b43bf4b4bf.png)
![begin mathsize 14px style b with rightwards arrow on top end style](https://images.topperlearning.com/topper/tinymce/cache/1ee88d50cc6a096c65659dea54046215.png)
![begin mathsize 14px style a with rightwards arrow on top end style](https://images.topperlearning.com/topper/tinymce/cache/d9ede463a364dc4ae9a804b43bf4b4bf.png)
![begin mathsize 14px style angle end style](https://images.topperlearning.com/topper/tinymce/cache/f1d45fc0daa19a85e39a8446cd27e48f.png)
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](https://images.topperlearning.com/topper/tinymce/cache/7cc00ac3f45ebe2a2f747b3eac2a3def.png)
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](https://images.topperlearning.com/topper/tinymce/cache/ab44416c438fefed1019a9a06d07724e.png)
![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](https://images.topperlearning.com/topper/tinymce/cache/e48070b128856e47bd4a1842eeea438e.png)
![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](https://images.topperlearning.com/topper/tinymce/cache/e5527fe160b944e52c13be0a345cc863.png)
Answered by Thiyagarajan K | 22 Feb, 2023, 12:12: PM
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