Answer
445.2k+ views
Hint – You can start the solution by drawing a well-labelled diagram with all the vectors ($A$,$B$and$R$) originating from a common point. The equations for the magnitude of the resultant vector and the direction of the resultant vector are $R = \sqrt {{A^2} + {B^2} + 2AB\operatorname{Cos} \theta } $ and $\tan \alpha = \dfrac{{B\sin \theta }}{{A + B\cos \theta }}$ respectively. Use the second equation given above to reach the solution.
Complete answer:
To solve this equation, consider the diagram given below
The arrangement of $A$, $B$ and $R$ (Resultant) vectors is done in such a way that it is easy to co-relate with the other two vectors.
We know,
$\angle AOB = \theta $,
And \[\angle ROA = \dfrac{\theta }{2}\]
We also know,
$\angle BOR = \angle AOB - \angle ROA$
$ \Rightarrow \angle BOR = \theta - \dfrac{\theta }{2}$
$ \Rightarrow \angle BOR = \dfrac{\theta }{2}$
The equation for the $\angle ROA$ is as follows –
$\tan \alpha = \dfrac{{B\sin \theta }}{{A + B\cos \theta }}$
$ \Rightarrow \dfrac{{\sin (\dfrac{\theta }{2})}}{{\cos (\dfrac{\theta }{2})}} = \dfrac{{2B(\dfrac{\theta }{2})\cos (\dfrac{\theta }{2})}}{{A + B\cos \theta }}$
$ \Rightarrow A + B\cos \theta = B{\cos ^2}\theta (\dfrac{\theta }{2})$
$ \Rightarrow A + B[2{\cos ^2}(\dfrac{\theta }{2}) - 1] = 2B{\cos ^2}(\dfrac{\theta }{2})$
$ \Rightarrow A = B$
Hence, Option C is the correct option
Additional Information:
A vector is a mathematical quantity that has both a magnitude (size) and a direction. To imagine what a vector is like, imagine asking someone for directions in an unknown area and they tell you, “Go $5km$ towards the West”. In this sentence, we see an example of a displacement vector, “\[5km\]” is the magnitude of the displacement vector and “towards the North” is the indicator of the direction of the displacement vector.
A vector quantity is different from a scalar quantity in the fact that a scalar quantity has only magnitude, but a vector quantity possesses both direction and magnitude. Unlike scalar quantities, vector quantities cannot undergo any mathematical operation, instead they undergo Dot product and Cross product.
Some examples of vectors are – Displacement, Force, Acceleration, Velocity, Momentum, etc.
Note – You can also get to the solution by not going through the mathematical calculations and just focusing on the theoretical part. You can make an argument that we know that $\angle BOR = \dfrac{\theta }{2}$ as , and. If the angles of the vector are the same with bothand. Then we can safely conclude that $A = B$, as only this condition can satisfy the given data.
Complete answer:
To solve this equation, consider the diagram given below
![seo images](https://www.vedantu.com/question-sets/945d4768-ab09-4a6f-a8a5-69ef7651cd448308547689781244142.png)
The arrangement of $A$, $B$ and $R$ (Resultant) vectors is done in such a way that it is easy to co-relate with the other two vectors.
We know,
$\angle AOB = \theta $,
And \[\angle ROA = \dfrac{\theta }{2}\]
We also know,
$\angle BOR = \angle AOB - \angle ROA$
$ \Rightarrow \angle BOR = \theta - \dfrac{\theta }{2}$
$ \Rightarrow \angle BOR = \dfrac{\theta }{2}$
The equation for the $\angle ROA$ is as follows –
$\tan \alpha = \dfrac{{B\sin \theta }}{{A + B\cos \theta }}$
$ \Rightarrow \dfrac{{\sin (\dfrac{\theta }{2})}}{{\cos (\dfrac{\theta }{2})}} = \dfrac{{2B(\dfrac{\theta }{2})\cos (\dfrac{\theta }{2})}}{{A + B\cos \theta }}$
$ \Rightarrow A + B\cos \theta = B{\cos ^2}\theta (\dfrac{\theta }{2})$
$ \Rightarrow A + B[2{\cos ^2}(\dfrac{\theta }{2}) - 1] = 2B{\cos ^2}(\dfrac{\theta }{2})$
$ \Rightarrow A = B$
Hence, Option C is the correct option
Additional Information:
A vector is a mathematical quantity that has both a magnitude (size) and a direction. To imagine what a vector is like, imagine asking someone for directions in an unknown area and they tell you, “Go $5km$ towards the West”. In this sentence, we see an example of a displacement vector, “\[5km\]” is the magnitude of the displacement vector and “towards the North” is the indicator of the direction of the displacement vector.
A vector quantity is different from a scalar quantity in the fact that a scalar quantity has only magnitude, but a vector quantity possesses both direction and magnitude. Unlike scalar quantities, vector quantities cannot undergo any mathematical operation, instead they undergo Dot product and Cross product.
Some examples of vectors are – Displacement, Force, Acceleration, Velocity, Momentum, etc.
Note – You can also get to the solution by not going through the mathematical calculations and just focusing on the theoretical part. You can make an argument that we know that $\angle BOR = \dfrac{\theta }{2}$ as , and. If the angles of the vector are the same with bothand. Then we can safely conclude that $A = B$, as only this condition can satisfy the given data.
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