Question

The angle between two vectors $A$ and $B$ is $\theta$ .Vector $R$ is the resultant of the two vectors. If $R$ makes an angle ${\displaystyle \frac{\theta }{2}}$ with $A$, then

A.)$A=2B$
B.)$A={\displaystyle \frac{B}{2}}$
C.)$A=B$
D.$AB=1$

Answer 1

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 $R=\sqrt{A^2+B^2+2AB\mathrm{Cos}\theta }$ and $\tan \alpha ={\displaystyle \frac{B\sin \theta }{A+B\cos \theta }}$ 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,

$\mathrm{\angle }AOB=\theta$,
And $$\mathrm{\angle }ROA={\displaystyle \frac{\theta }{2}}$$

We also know,

$\mathrm{\angle }BOR=\mathrm{\angle }AOB-\mathrm{\angle }ROA$
$\Rightarrow \mathrm{\angle }BOR=\theta -{\displaystyle \frac{\theta }{2}}$
$\Rightarrow \mathrm{\angle }BOR={\displaystyle \frac{\theta }{2}}$

The equation for the $\mathrm{\angle }ROA$ is as follows –

$\tan \alpha ={\displaystyle \frac{B\sin \theta }{A+B\cos \theta }}$
$\Rightarrow {\displaystyle \frac{\sin ({\displaystyle \frac{\theta }{2}})}{\cos ({\displaystyle \frac{\theta }{2}})}}={\displaystyle \frac{2B({\displaystyle \frac{\theta }{2}})\cos ({\displaystyle \frac{\theta }{2}})}{A+B\cos \theta }}$
$\Rightarrow A+B\cos \theta =B{\cos }^2\theta ({\displaystyle \frac{\theta }{2}})$
$\Rightarrow A+B[2{\cos }^2({\displaystyle \frac{\theta }{2}})-1]=2B{\cos }^2({\displaystyle \frac{\theta }{2}})$
$\Rightarrow A=B$

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 $\mathrm{\angle }BOR={\displaystyle \frac{\theta }{2}}$ 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.