The resultant of two forces $3P$ and $2P$ is $R$. If the first force is doubled then the resultant is also doubled. The angle between the two forces is  ........... $^o$

If the angle between $\hat a$ and $\hat b$ is $60^o$, then which of the following  vector $(s)$ have magnitude one

$(A)$ $\frac{\hat a + \hat b}{\sqrt 3}$     $(B)$ $\hat a + \widehat b$     $(C)$ $\hat a$      $(D)$ $\hat b$

Assertion $A$ : If $A, B, C, D$ are four points on a semi-circular arc with centre at $'O'$ such that $|\overrightarrow{{AB}}|=|\overrightarrow{{BC}}|=|\overrightarrow{{CD}}|$, then $\overrightarrow{{AB}}+\overrightarrow{{AC}}+\overrightarrow{{AD}}=4 \overrightarrow{{AO}}+\overrightarrow{{OB}}+\overrightarrow{{OC}}$

Reason $R$ : Polygon law of vector addition yields $\overrightarrow{A B}+\overrightarrow{B C}+\overrightarrow{C D}+\overrightarrow{A D}=2 \overrightarrow{A O}$

In the light of the above statements, choose the most appropriate answer from the options given below

Given that $\overrightarrow A + \overrightarrow B + \overrightarrow C= 0$ out of three vectors two are equal in magnitude and the magnitude of third vector is $\sqrt 2 $ times that of either of the two having equal magnitude. Then the angles between vectors are given by

If $\left| {{{\vec v}_1} + {{\vec v}_2}} \right| = \left| {{{\vec v}_1} - {{\vec v}_2}} \right|$ and ${{{\vec v}_1}}$ and ${{{\vec v}_2}}$ are finite, then