Answer
64.8k+ views
Hint: First of all, write the given quantities. For finding the magnitude of the change in the position vector, draw the relevant figure for sake of clear understanding and better understanding. Now, using this figure and given quantities, find out the change in magnitude in the position vector.
Complete step by step solution:
Given: ${{l}}$= length of the vector
And ${{\theta }}$ = angle of inclination
![](https://www.vedantu.com/question-sets/a01b8406-99f5-4967-b37e-00214f553c072605782671244306472.png)
Two vectors represented in both magnitude and direction by two adjacent sides of a parallelogram drawn from a common point, then their resultant is completely represented both in magnitude and direction by the diagonal of the parallelogram passing through that point.
Resultant vector is given by
$\Delta {{\vec r = r}}_2^2 - {{r}}_1^2$
According to the question, ${{{r}}_{{1}}}{{ = }}{{{r}}_{{2}}}{{ = l}}$
Now using parallelogram law of vector addition,
$\Delta {{r = }}\sqrt {{{{r}}_{{1}}}^{{2}}{{ + }}{{{r}}_{{2}}}^{{2}}{{ - 2}}{{{r}}_{{1}}}{{{r}}_{{2}}}{{cos\theta }}} $
Now, on substituting the value of${{{r}}_{{1}}}{{ and }}{{{r}}_{{2}}}$, we get
$\Delta {{r = }}\sqrt {{{{l}}^{{2}}}{{ + }}{{{l}}^{{2}}}{{ - 2}}{{{l}}^{{2}}}{{cos\theta }}} $
On further simplification, we get
$\Rightarrow \Delta {{r = }}\sqrt {{{2}}{{{l}}^{{2}}}{{(1 - cos\theta )}}} \\
\Rightarrow \Delta {{r = l}}\sqrt {{{2(2sin}}\dfrac{{{\theta }}}{{{2}}}{{)}}} \\
\therefore {{ \Delta r = 2lsin}}\left( {\dfrac{{{\theta }}}{{{2}}}} \right) $
Therefore, option (B) is the correct choice.
Note: A vector which gives the position of a point with reference to the origin of the coordinate system, is called position vector. Scalars can be added algebraically while the vectors don't because of the fact that vectors possess both the magnitude as well as direction. While adding two vectors with the help of parallelogram vector addition rule, we have to ensure that the two vectors act either towards a point or away from a point. However, the magnitude of resultant of two vectors is minimum when they act in opposite directions and the magnitude of the resultant of two vectors is maximum when they act in the same direction.
Complete step by step solution:
Given: ${{l}}$= length of the vector
And ${{\theta }}$ = angle of inclination
![](https://www.vedantu.com/question-sets/a01b8406-99f5-4967-b37e-00214f553c072605782671244306472.png)
Two vectors represented in both magnitude and direction by two adjacent sides of a parallelogram drawn from a common point, then their resultant is completely represented both in magnitude and direction by the diagonal of the parallelogram passing through that point.
Resultant vector is given by
$\Delta {{\vec r = r}}_2^2 - {{r}}_1^2$
According to the question, ${{{r}}_{{1}}}{{ = }}{{{r}}_{{2}}}{{ = l}}$
Now using parallelogram law of vector addition,
$\Delta {{r = }}\sqrt {{{{r}}_{{1}}}^{{2}}{{ + }}{{{r}}_{{2}}}^{{2}}{{ - 2}}{{{r}}_{{1}}}{{{r}}_{{2}}}{{cos\theta }}} $
Now, on substituting the value of${{{r}}_{{1}}}{{ and }}{{{r}}_{{2}}}$, we get
$\Delta {{r = }}\sqrt {{{{l}}^{{2}}}{{ + }}{{{l}}^{{2}}}{{ - 2}}{{{l}}^{{2}}}{{cos\theta }}} $
On further simplification, we get
$\Rightarrow \Delta {{r = }}\sqrt {{{2}}{{{l}}^{{2}}}{{(1 - cos\theta )}}} \\
\Rightarrow \Delta {{r = l}}\sqrt {{{2(2sin}}\dfrac{{{\theta }}}{{{2}}}{{)}}} \\
\therefore {{ \Delta r = 2lsin}}\left( {\dfrac{{{\theta }}}{{{2}}}} \right) $
Therefore, option (B) is the correct choice.
Note: A vector which gives the position of a point with reference to the origin of the coordinate system, is called position vector. Scalars can be added algebraically while the vectors don't because of the fact that vectors possess both the magnitude as well as direction. While adding two vectors with the help of parallelogram vector addition rule, we have to ensure that the two vectors act either towards a point or away from a point. However, the magnitude of resultant of two vectors is minimum when they act in opposite directions and the magnitude of the resultant of two vectors is maximum when they act in the same direction.
Recently Updated Pages
Write a composition in approximately 450 500 words class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Arrange the sentences P Q R between S1 and S5 such class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What is the common property of the oxides CONO and class 10 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What happens when dilute hydrochloric acid is added class 10 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
If four points A63B 35C4 2 and Dx3x are given in such class 10 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
The area of square inscribed in a circle of diameter class 10 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Other Pages
Excluding stoppages the speed of a bus is 54 kmph and class 11 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
In the ground state an element has 13 electrons in class 11 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Electric field due to uniformly charged sphere class 12 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
A boat takes 2 hours to go 8 km and come back to a class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
According to classical free electron theory A There class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)